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SYMBOLIC DYNAMICS FOR THREE DIMENSIONAL FLOWS

WITH POSITIVE TOPOLOGICAL ENTROPY

YURI LIMA AND OMRI M. SARIG

Abstract. We construct symbolic dynamics on sets of full measure (w.r.t.an ergodic measure of positive entropy) for C1+ε flows on compact smooth

three-dimensional manifolds. One consequence is that the geodesic flow on

the unit tangent bundle of a compact C∞ surface has at least const×(ehT /T )simple closed orbits of period less than T , whenever the topological entropy h

is positive — and without further assumptions on the curvature.

1. Introduction

The aim of this paper is to develop symbolic dynamics for smooth flows withtopological entropy h > 0, on three-dimensional compact Riemannian manifolds.

Earlier works treated geodesic flows on hyperbolic surfaces, geodesic flows onsurfaces with variable negative curvature, and uniformly hyperbolic flows in anydimension [Ser81, Ser87, KU07],[Rat69],[Rat73, Bow73]. This work only assumesthat h > 0 and that the flow has positive speed (i.e. the vector field which generatesit has no zeroes).

This generality allows us to cover several cases of interest that could not betreated before, for example:

(1) Geodesic flows with positive entropy in positive curvature: There are many Rie-mannian metrics with positive curvature somewhere (even everywhere) whosegeodesic flow has positive topological entropy [Don88, BG89, KW02, CBP02].

(2) Reeb flows with positive entropy: These arise from Hamiltonian flows on sur-faces of constant energy, see [Hut10]. Examples with positive topological en-tropy are given in [MS11]. (This application was suggested to us by G. Forni.)

(3) Abstract non-uniformly hyperbolic flows in three dimensions, as in [BP07],[Pes76].

The statement of our main result is somewhat technical, therefore we begin witha down-to-earth corollary. Suppose ϕ is a flow. A simple closed orbit of length `is a parameterized curve γ(t) = ϕt(p), 0 ≤ t ≤ ` s.t. γ(0) = γ(`) and γ(0) 6= γ(t)when 0 < t < `. The trace of γ is the set γ(t) : 0 ≤ t ≤ `. Let [γ] denote theequivalence class of the relation γ1 ∼ γ2 ⇔ γ1, γ2 have equal lengths and traces.Let π(T ) := #[γ] : `(γ) ≤ T, γ is simple.Theorem 1.1. Suppose ϕ is a C∞ flow with positive speed on a C∞ compactthree dimensional manifold. If ϕ has positive topological entropy h, then there is apositive constant C s.t. π(T ) ≥ C(ehT /T ) for all T large enough.

Date: July 24, 2014.2010 Mathematics Subject Classification. 37B10,37C10 (primary), 37C35 (secondary).Key words and phrases. Markov partitions, symbolic dynamics, geodesic flows.Y.L. was partially supported by the Brin Fellowship. O.S. was partially supported by ERC

award ERC-2009-StG n 239885.

1

2 YURI LIMA AND OMRI M. SARIG

The theorem strengthens Katok’s bound lim infT→∞1T log π(T ) ≥ h, see [Kat80,

Kat82]. It extends to flows of lesser regularity, under the additional assumptionthat they possess a measure of maximal entropy (Theorem 8.1). The lower bound

C( eht

T ) is sharp in many special cases, but not in the general setup of this paper,see §8 for a discussion.

We obtain Theorem 1.1 from a “change of coordinates” that transforms ϕ intoa “symbolic flow” whose orbits are easier to understand. We proceed to define themodeling flow. Let G be a directed graph with countable set of vertices V . Wewrite v → w if there is an edge from v to w, and assume throughout that for everyv there are u,w s.t. u→ v, v → w.

Topological Markov shifts: The topological Markov shift associated to G isthe discrete-time topological dynamical system σ : Σ→ Σ where

Σ = Σ(G ) := paths on G = vii∈Z : vi → vi+1 for all i ∈ Z,

equipped with the metric d(v, w) := exp[−min|n| : vn 6= wn], and σ : Σ → Σ isthe left shift map σ : vii∈Z 7→ vi+1i∈Z.

Birkhoff cocycle: Suppose r : Σ → R is a function. The Birkhoff sums of rare rn := r + r σ + · · · + r σn−1 (n ≥ 1). There is a unique way to extend thedefinition to n ≤ 0 in such a way that the cocycle identity rm+n = rn + rm σnholds for all m,n ∈ Z: r0 := 0 and rn := −r|n| σ−|n| (n < 0).

Topological Markov flow: Suppose r : Σ → R+ is Holder continuous andbounded away from zero and infinity. The topological Markov flow with roof functionr and base map σ : Σ→ Σ is the flow σr : Σr → Σr where

Σr := (v, t) : v ∈ Σ, 0 ≤ t < r(v) , στr (v, t) = (σn(v), t+ τ − rn(v))

for the unique n ∈ Z s.t. 0 ≤ t+ τ − rn(v) < r(σn(v)).Informally, σr increases the t coordinate at unit speed subject to the identifica-

tions (v, r(v)) ∼ (σ(v), 0). The cocycle identity guarantees that στ1+τ2r = στ1r στ2r .

Regular part: Σ# := v ∈ Σ : vii≤0, vii≥0 have constant subsequences iscalled the regular part of Σ. Σ#

r := (v, t) ∈ Σr : v ∈ Σ# is called the regularpart of Σr. By Poincare’s Recurrence Theorem, Σ#

r has full measure w.r.t. to anyσr–invariant probability measure, and it contains all the closed orbits of σr.

Here is our main result: Let M be a 3–dimensional compact C∞ manifold, andX be a C1+β (0 < β < 1) vector field on M s.t. Xp 6= 0 for all p. Let ϕ : M →Mbe the flow determined by X, and let µ be a ϕ–invariant Borel probability measure.

Theorem 1.2. If µ is ergodic and the entropy of µ is positive, then there is atopological Markov flow σr : Σr → Σr and a map πr : Σr →M s.t.:

(1) r : Σ→ R+ is Holder continuous and bounded away from zero and infinity.(2) πr is Holder continuous with respect to the Bowen-Walters metric (see §5).(3) πr σtr = ϕt πr for all t ∈ R.(4) πr[Σ

#r ] has full measure with respect to µ.

(5) If p = πr(x, t) where xi = v for infinitely many i < 0 and xi = w for infinitelymany i > 0, then #(y, s) ∈ Σ#

r : πr(y, s) = p ≤ N(v, w) <∞.

(6) ∃N = N(µ) <∞ s.t. µ–a.e. p ∈M has exactly N pre-images in Σ#r .

SYMBOLIC DYNAMICS FOR FLOWS 3

Some of the applications we have in mind require a version of this result for non-ergodic measures. To state it, we need to recall some facts from smooth ergodictheory [BP07]. Let TpM be the tangent space at p and let (dϕt)p : TpM → Tϕt(p)Mbe the differential of ϕt : M →M at p. Suppose µ is a ϕ–invariant Borel probabilitymeasure on M (not necessarily ergodic). By the Oseledets Theorem, for µ–a.e.p ∈ M , for every 0 6= ~v ∈ TpM , the limit χp(~v) := limt→∞

1t log ‖(dϕt)p~v‖ϕt(p)

exists. The values of χp(·) are called the Lyapunov exponents at p. If dim(M) = 3,then there are at most three such values. At least one of them, χp(Xp), equals zero.

Hyperbolic measures: Suppose χ0 > 0. A χ0–hyperbolic measure is an invariantmeasure µ s.t. µ–a.e. p ∈ M has one Lyapunov exponent in (−∞,−χ0), oneLyapunov exponent in (χ0,∞) and one Lyapunov exponent equal to zero.

In dimension 3, every ergodic invariant measure with positive metric entropy isχ0–hyperbolic, for any 0 < χ0 < hµ(ϕ) [Rue78]. But some hyperbolic measures,e.g. those carried by hyperbolic closed orbits, have zero entropy.

Theorem 1.3. Suppose µ is a χ0–hyperbolic invariant probability measure for someχ0 > 0. Then there are a topological Markov flow σr : Σr → Σr and a mapπr : Σr →M satisfying (1)–(5) in Theorem 1.2. If µ is ergodic, (6) holds as well.

Results in this spirit were first proved by Ratner and Bowen for Anosov flowsand Axiom A flows in any dimension [Rat69, Rat73, Bow73], using the techniqueof Markov partitions introduced by Adler & Weiss and Sinai for discrete-time dy-namical systems [AW67, AW70, Sin68a, Sin68b].1

In 1975 Bowen gave a new construction of Markov partitions for Axiom A diffeo-morphisms, using shadowing techniques [Bow75, Bow78]. One of us extended thesetechniques to general C1+β surface diffeomorphisms [Sar13]. Our strategy is toapply these methods to a suitable Poincare section for the flow. The main difficultyis that [Sar13] deals with diffeomorphisms, but Poincare sections are discontinuous.

In part 1 of the paper, we construct a Poincare section Λ with the followingproperty: If f : Λ → Λ is the section map and S ⊂ Λ is the set of discontinuitiesof f , then lim inf |n|→∞

1n log distΛ(fn(p),S) = 0 a.e. in Λ. This places us in the

context of “non-uniformly hyperbolic maps with singularities” studied in [KSLP86].In part 2 we explain why the methods of [Sar13] apply to f : Λ→ Λ despite its

discontinuities. The result is a countable Markov partition for f : Λ → Λ, whichleads to a coding of f as a topological Markov shift, and a coding of ϕ : M → Mas a topological Markov flow.

In part 3, we provide two applications: Theorem 1.1 on the growth of the numberof closed orbits; and a result saying that the set of measures of maximal entropy isfinite or countable. The proof of Theorem 1.1 uses a mixing/constant suspensiondichotomy for topological Markov flows, in the spirit of [Pla72].

Standing assumptions. M is a compact 3–dimensional C∞ Riemannian manifoldwithout boundary, with tangent bundle TM =

⋃p∈M TpM , Riemannian metric

〈·, ·〉p, norm ‖ ·‖p, and exponential map expp (this is different from the Expp in §3).

Given Y ⊂ M , distY (y1, y2) := inflengths of rectifiable curves γ ⊂ Y from y1

to y2, where inf ∅ := ∞. Given two metric spaces (A, dA),(B, dB) and a map

F : A→ B, Holα(F ) := supx 6=ydB(F (x),F (y))dA(x,y)α , and Lip(F ) := Hol1(F ).

1For geodesic flows on hyperbolic surfaces, alternative geometric and number theoretic methods

are possible. See [Ser81, Ser91, KU07] and references therein.


X : M → TM is a C1+β vector field on M (0 < β < 1), and ϕ : M → M isthe flow generated by X. This means that ϕ is a one-parameter family of mapsϕt : M → M s.t. ϕt+s = ϕt ϕs for all t, s ∈ R, and s.t. Xp(f) = d

dt

∣∣t=0

f [ϕt(p)]

for all f ∈ C∞(M). In this case (t, p) 7→ ϕt(p) is a C1+β map [−1, 1] ×M → M[EM70, page 112]. We assume throughout that Xp 6= 0 for all p.

Part 1. The Poincare section

2. Poincare sections

Basic definitions. Suppose ϕ : M →M is a flow.

Poincare section: Λ ⊂ M Borel s.t. for every p ∈ M , t > 0 : ϕt(p) ∈ Λ is asequence tending to +∞, and t < 0 : ϕt(p) ∈ Λ is a sequence tending to −∞.

Roof function: RΛ : Λ→ (0,∞), RΛ(p) := mint > 0 : ϕt(p) ∈ Λ.

Poincare map: fΛ : Λ→ Λ, fΛ(p) := ϕRΛ(p)(p).

Induced measure: Every ϕ–invariant probability measure µ on M induces an fΛ–

invariant measure µΛ on Λ s.t.∫Mgdµ = 1∫

ΛRΛdµΛ

∫Λ

(∫ RΛ(p)

0g[ϕt(p)]dt

)dµΛ(p)

for all g ∈ L1(µ).

Uniform Poincare section: Λ is called uniform, if its roof function is boundedaway from zero and infinity. If Λ is uniform, then µΛ is finite, and can be normalized.With this normalization, for every Borel subset E ⊂ Λ and 0 < ε < inf RΛ,µΛ(E) = µ[

⋃0<t<ε ϕ

t(E)]/µ[⋃

0<t<ε ϕt(Λ)].

All the Poincare sections in this paper will be uniform, and they will all be finitedisjoint unions of connected embedded smooth submanifolds with boundary. Let∂Λ denote the union of the boundaries of these submanifolds. ∂Λ will introducediscontinuities to the Poincare map of Λ.

Singular set: The singular set of a Poincare section Λ is

S(Λ) :=

p ∈ Λ :p does not have a relative neighborhood V ⊂ Λ \ ∂Λ s.t.V is diffeomorphic to an open disc, and fΛ : V → fΛ(V )and f−1

Λ : V → f−1Λ (V ) are diffeomorphisms

.

Regular set: Λ′ := Λ \S(Λ).

Basic constructions. Let ϕ be a flow satisfying our standing assumptions.

Canonical transverse disc: Sr(p) := expp(~v) : ~v ∈ TpM,~v ⊥ Xp, ‖~v‖p ≤ r.

Canonical flow box: FBr(p) := ϕt(q) : q ∈ Sr(p), |t| ≤ r.

The following lemmas are standard, see the appendix for proofs.

Lemma 2.1. There is a constant rs > 0 which only depends on M and ϕ s.t.for every p ∈ M and 0 < r < rs, S := Sr(p) is a C∞ embedded closed disc,|](Xq, TqS)| ≥ 1

2 radians for all q ∈ S, and distM (·, ·) ≤ distS(·, ·) ≤ 2 distM (·, ·).

Lemma 2.2. There are constants rf , d ∈ (0, 1) which only depend on M and ϕ s.t.for every p ∈ M , FBrf (p) contains an open ball with center p and radius d, and(q, t) 7→ ϕt(q) is a diffeomorphism from Srf (p)× [−rf , rf ] onto FBrf (p).


Lemma 2.3. There are constants L,H > 1 which only depend on M and ϕ s.t.tp : FBrf (p) → [−rf , rf ] and qp : FBrf (p) → Srf (p) defined by z = ϕtp(z)[qp(z)]are well-defined maps s.t. Lip(tp),Lip(qp) ≤ L and ‖tp‖C1+β , ‖qp‖C1+β ≤ H.

tp and qp are called the flow box coordinates.Set r := 10−1 min1, rs, rf , d/(1 + max ‖Xp‖).

Standard Poincare section: A Poincare section of the form

Λ = Λ(p1, . . . , pN ; r) :=

N⊎i=1

Sr(pi)

where r < r, supRΛ < r, and Sr(pi) are pairwise disjoint. The points p1, . . . , pNare called the centers of Λ, and r is called the radius of Λ.

We will discuss the existence of standard Poincare sections in the next subsection.Let us assume for the moment that they exist, and study some of their properties.Fix a standard Poincare section Λ = Λ(p1, . . . , pN ; r) and write f = fΛ, R = RΛ,S := S(Λ), and Λ′ := Λ \S.

Lemma 2.4. Every standard Poincare section is uniform: inf R > 0.

Proof. Let x ∈ Sr(pi), f(x) ∈ Sr(pj). If i = j then R(x) > 2rf , because ϕt(x) mustleave FBrf (pi) before it can return to it. If i 6= j, then ϕt(x)0≤t≤R(x) is a curvefrom Sr(pi) to Sr(pj), and R(x) ≥ distM (Sr(pi), Sr(pj))/max ‖Xp‖.

Lemma 2.5. R, f and f−1 are differentiable on Λ′, and ∃C only depending onM and ϕ s.t. supx∈Λ′ ‖dRx‖ < C, supx∈Λ′ ‖dfx‖ < C, supx∈Λ′ ‖(dfx)−1‖ < C,‖f |U‖C1+β < C and ‖f−1|U‖C1+β < C for all open and connected U ⊂ Λ′.

Proof. Suppose x ∈ Λ′, then ∃i, j, k s.t. f−1(x) ∈ Sr(pi), x ∈ Sr(pj), and f(x) ∈Sr(pk). Since f is continuous at x and the canonical discs which form Λ are closedand disjoint, x has an open neighborhood V in Sr(pj), s.t. for all y ∈ V , f(y) ∈Sr(pk) and f−1(y) ∈ Sr(pi).

Since supR < r < 10−1d/max ‖Xp‖, for every y ∈ V ,

distM (y, pk) ≤ distM (y, f(y)) + distM (f(y), pk) ≤ max ‖Xp‖ supR+ r < d.

Similarly, distM (y, pi) < d. Thus V ⊂ Bd(pi) ∩ Bd(pk) ⊂ FBrf (pi) ∩ FBrf (pk),

whence R|V = −tpk , f |V = qpk , f−1|V = qpi . Now use Lemma 2.3.

Let µ be a flow invariant probability measure, and let µΛ be the induced mea-sure on Λ. If µΛ(S) = 0, then µΛ[

⋃n∈Z f

n(S)] = 0, and the derivative cocycle

dfnx : TxΛ → Tfn(x)Λ is well-defined µΛ–a.e. By Lemma 2.5, log ‖dfx‖, log ‖df−1x ‖

are integrable (even bounded), so the Oseledets Multiplicative Ergodic Theoremapplies, and f has well-defined Lyapunov exponents µΛ–a.e. Fix χ > 0.

Lemma 2.6. Suppose µΛ(S) = 0. If ϕ has one Lyapunov exponent in (−∞,−χ)and one Lyapunov exponent in (χ,∞) µ–a.e., then f has one Lyapunov exponentin (−| lnC|,−χ inf R) and one Lyapunov exponent in (χ inf R, | lnC|) µΛ–a.e.

Proof. Let Ωχ denote the set of points where the flow has one zero Lyapunovexponent, one Lyapunov exponent in (−∞,−χ) and one Lyapunov exponent in(χ,∞). By assumption µ[Ωcχ] = 0. Thus Λχ := x ∈ Λ \

⋃n∈Z f

−n(S) : ∃t >0 s.t. ϕt(x) ∈ Ωχ has full measure with respect to µΛ.


Let Λ∗χ := x ∈ Λχ : χ(x,~v) := limn→±∞

1n log ‖dfnx ~v‖ exists for all 0 6= ~v ∈ TxΛ.

By Oseledets’ theorem, Λ∗χ has full µΛ–measure. By Lemma 2.5, |χ(x,~v)| ≤ | lnC|.The Lyapunov exponents of ϕ are constant along flow lines, therefore for every

x ∈ Λ∗χ, there are vectors esx, eux ∈ TxM s.t. lim

t→∞1t log ‖dϕtxesx‖ϕt(x) < −χ and

limt→∞

1t log ‖dϕtxeux‖ϕt(x) > χ. Let ~n(x) := Xx

‖Xx‖ . Since limt→∞

1t log ‖dϕtx~n(x)‖ϕt(x) =

0, esx, eux, ~n(x) span TxM . The vectors esx, eux are not necessarily in TxΛ.

Pick two independent vectors ~v1, ~v2 ∈ TxΛ and write ~vi = αiesx + βie

ux + γi~n(x)

(i = 1, 2).(α1

β1

),(α2

β2

)must be linearly independent, otherwise some non-trivial linear

combination of ~v1, ~v2 equals ~n(x), which is impossible since span~v1, ~v2 = TxΛ andΛ is tranverse to the flow. It follows that TxΛ contains two vectors of the form

~vsx = esx + γs~n(x) , ~vux = eux + γu~n(x).

These vectors are the projections of esx, eux to TxΛ along ~n(x). We will estimate

their Lyapunov exponents.Write Λ = Λ(p1, . . . , pN ; r). As in the proof of Lemma 2.5, for every x ∈ Λ \S,

if f(x) ∈ Sr(pi), then x has a neighborhood V in Λ s.t. V ⊂ FBrf (pi), R|V = −tpi ,and f |V = qpi . More generally, suppose fn(x) ∈ Sr(qn) for qn ∈ p1, . . . , pN. Ifx 6∈

⋃k∈Z f

k(S), then there are open neighborhoods Vn of x in Λ s.t.

fn−1(Vn) ∈ FBrf (qn), and fn|Vn = (qqn · · · qq1)|Vn . (2.1)

By the definition of the flow box coordinates, for every i, qqi(·) = ϕ−tqi (·)(·). Sincex 6∈

⋃k∈Z f

k(S), tqi is continuous on a neighborhood of f i−1(x), therefore the

smaller Vn, the closer −tqi |fi−1(Vn) is to R(f i−1(x)). If Vn is small enough, and

Rn := R(x) +R(f(x)) + · · ·+R(fn−1(x)),

then ϕRn(y) ∈ FBrf (qn) for all y ∈ Vn, and we can decompose

(qqn · · · qq1)(y) = (qqn ϕRn)(y) (y ∈ Vn). (2.2)

We emphasize that the power Rn is the same for all y ∈ Vn.We use (2.1), (2.2) to calculate dfnx ~v

sx. First note that (dqq1)x~n(x) = ~0: let

γ(t) = ϕt(x), then qq1 [γ(t)] = qq1(x) for all |t| small, so ddt

∣∣t=0

qq1 [γ(t)] = ~0. By

(2.1), dfnx ~vsx = d(qqn· · ·qq1)esx. By (2.2), ‖dfnx ~vsx‖ ≤ maxi ‖dqqi‖·‖dϕRnx esx‖ϕRn (x),

whence lim supn→∞1n log ‖dfnx ~vsx‖fn(x) ≤ −χ lim supn→∞

Rnn ≤ −χ inf R.

Applying this argument to the reverse flow ψt := ϕ−t, we find that the otherLyapunov exponent belongs to (χ inf R,∞).

Remark. If µ is ergodic, then lim supn→∞Rnn =

∫RdµΛ = 1, and we get the

stronger estimate that the Lyapunov exponents of f are a.s. outside (−χ, χ).

Adapted Poincare sections. Let Λ be a standard Poincare section. Let distΛ de-note the intrinsic Riemannian distance on Λ (with the convention that the distancebetween different connected components of Λ is infinite). Let µ be a ϕ–invariantprobability measure, and let µΛ be the induced probability measure on Λ.

Recall that fΛ : Λ → Λ may have singularities. The following definition ismotivated by the treatment of Pesin theory for maps with singularities in [KSLP86].

Adapted Poincare section: A standard Poincare section Λ is adapted to µ, if

(1) µΛ(S) = 0, where S = S(Λ) is the singular set of Λ,(2) lim

n→∞1n log distΛ(fnΛ(p),S) = 0 µΛ–a.e.,


(3) limn→∞

1n log distΛ(f−nΛ (p),S) = 0 µΛ–a.e.

Notice that (2)⇒ (1), by Poincare’s recurrence theorem.

We wish to show that any ϕ–invariant Borel probability measure has adaptedPoincare sections. The idea is to construct a one-parameter family of standardPoincare sections Λr, and show that Λr is adapted to µ for a.e. r. The family isconstructed in the next lemma:

Lemma 2.7. For every h0 > 0,K0 > 1 there are p1, . . . , pN ∈M , 0 < ρ0 < h0/K0

s.t. for every r ∈ [ρ0,K0ρ0], Λ(p1, . . . , pN ; r) is a standard Poincare section withroof function and radius bounded above by h0.

Proof. By compactness, M can be covered by a finite number of flow boxes FBr(zi)with radius r. The union of Sr(zi) is a Poincare section, but this section is notnecessarily standard, because Sr(zi) are not necessarily pairwise disjoint.

To solve this problem we approximate each Sr(zi) by a finite “net” of points zijk,

and shift each zijk up or down with the flow to points pijk = ϕθijk(zijk) in such a

way that SR0(pijk) are pairwise disjoint for some R0 < r which is still large enough

to ensure that⋃SR0

(zijk) is a Poincare section. The details are somewhat lengthyto write down, and they will not be used elsewhere in the paper. The reader whobelieves that this can be done, may skip the proof.

We begin with the choice of some constants. Let:

h0 > 0 small, K0 > 1 large (given to us). W.l.o.g., 0 < h0 < rf .

rinj ∈ (0, 1) s.t. expp : ~v ∈ TpM : ‖~v‖ ≤ rinj → M is√

2–bi-Lipschitz for allp ∈M .

S0 := 1 + max ‖Xp‖ and r, d,L are as in Lemmas 2.1–2.3. Recall that r, d ∈ (0, 1)and L > 1. r0 := 1

9 rdh0rinj/(K0 + S0). Notice that r0 <19 r,

19d,

19h0,

19rinj.

By Lemma 2.2 and the compactness of M , it is possible to cover M by finitelymany flow boxes FBr0(z1), . . . , FBr0(zN ). With this N in mind, let:

ρ0 := r0(10K0S0NL)−20. This is smaller than r0. R0 := K0ρ0. This is larger than ρ0, but still much smaller than r0. δ0 := ρ0/(8L

2). This is much smaller than r0. κ0 := d102K0L

4e, a big integer.

For every i, complete ~ni := Xzi/‖Xzi‖ to an orthonormal basis ~ui, ~vi, ~ni ofTziM , and let Ji : R2 →M be the map

Ji(x, y) = expzi(x~ui + y~vi),

then Sr0(zi) = Ji((x, y) : x2 + y2 ≤ r2

0). Ji is

√2–bi-Lipschitz, because r0 < rinj.

Let I := (j, k) ∈ Z2 : (jδ0)2 +(kδ0)2 ≤ r20. Given 1 ≤ i ≤ N and (j, k) ∈ I, define

zijk := Ji(jδ0, kδ0).

Then zijk : (j, k) ∈ I is a net of points in Sr0(zi), and for all (j, k) 6= (`,m)

1√2≤

distM (zijk, zi`m)

δ0√

(j − l)2 + (k −m)2≤√

2. (2.3)

We will construct points pijk := ϕθijk(zijk) with θijk ∈ [−r0, r0] s.t. SR0(pijk) are

pairwise disjoint. The following claim will help us prove disjointness.


Claim. Suppose pijk = ϕθijk(zijk), pi`m = ϕθ

i`m(zi`m), where θijk, θ

i`m ∈ [−r0, r0]. If

SR0(pijk)∩SR0

(ϕτ1(zγαβ)) 6= ∅ and SR0(pi`m)∩SR0

(ϕτ2(zγαβ)) 6= ∅ for the same zγαβand some τ1, τ2 ∈ [−r0, r0], then max|j − `|, |k −m| < κ0.

In particular, SR0(pijk) ∩ SR0

(pi`m) 6= ∅⇒ max|j − `|, |k −m| < κ0 (take zγαβ =

zi`m, τ1 = τ2 = θi`m).

Proof. SR0(pijk), SR0

(pi`m), zγαβ , ϕτ1(zγαβ), ϕτ2(zγαβ) are all contained in Bd(zi):

SR0(pijk) ⊂ Bd(zi), because if q ∈ SR0(pijk), then distM (q, zi) ≤ distM (q, pijk) +

distM (pijk, zijk) + distM (zijk, zi) ≤ R0 + r0S0 + r0 < d. Similarly, SR0

(pi`m) ⊂Bd(zi). zγαβ ∈ Bd(zi): distM (zγαβ , zi) ≤ distM (zγαβ , ϕ

τ1(zγαβ)) + distM (ϕτ1(zγαβ), pijk) +

distM (pijk, zijk) + distM (zijk, zi) ≤ r0S0 + 2R0 + r0S0 + r0 < d.

ϕτ1(zγαβ) ∈ Bd(zi): distM (ϕτ1(zγαβ), zi) ≤ distM (ϕτ1(zγαβ), pijk)+distM (pijk, zijk)+

distM (zijk, zi) < 2R0 + r0S0 + r0 < d. Similarly, ϕτ2(zγαβ) ∈ Bd(zi).

By Lemma 2.2, the flow box coordinates tzi(·), qzi(·) of ϕτ1(zγαβ), ϕτ2(zγαβ), zγαβ ,

and of every point in SR0(pijk), SR0(pi`m) are well-defined.Recall that tzi , qzi have Lipschitz constants less than L. In the set of circum-

stances we consider distM (pijk, ϕτ1(zγαβ)) ≤ 2R0 and qzi(p

ijk) = zijk, so

distM (zijk, qzi(zγαβ)) = distM (qzi(p

ijk), qzi(ϕ

τ1(zγαβ))) ≤ 2LR0.

Similarly, dist(zi`m, qzi(zγαβ)) ≤ 2LR0. It follows that distM (zijk, z

i`m) ≤ 4LR0. By

(2.3), max|j − `|, |k −m| ≤ 4√

2LR0/δ0 = 4√

2LK0ρ0

/(ρ0/8L

2) < κ0.

The claim is proved. We proceed to construct by induction θijk ∈ [−r0, r0] and

pijk := ϕθijk(zijk) such that SR0

(pijk) : i = 1, . . . , N ; (j, k) ∈ I are pairwise disjoint.

Basis of induction: ∃θ1jk ∈ [−r0, r0] s.t. SR0(p1

jk)(j,k)∈I are pairwise disjoint.

Construction: Let σ : 0, 1, . . . , κ0 − 1 × 0, 1, . . . , κ0 − 1 → 1, . . . , κ20 be a

bijection, and set σjk := σ(j modκ0, k modκ0

). This has the effect that

0 < max|j − `|, |k −m| < κ0 =⇒ |σjk − σ`m| ≥ 1.

We let θ1jk := 2R0Lσjk and p1

jk := ϕθ1jk(z1

jk). It is easy to check that 0 < θ1jk < r0.

One shows as in the proof of the claim that SR0(p1jk) ⊂ Bd(z1), therefore tz1 is

well-defined on SR0(p1jk). Since Lip(tz1) ≤ L and tz1(p1

jk) = θ1jk,

tz1[SR0(p1

jk)]⊂(θ1jk − LR0, θ

1jk + LR0

). (2.4)

Now suppose (j, k) 6= (`,m). If max|j − `|, |k − m| ≥ κ0, then SR0(p1jk) ∩

SR0(p1`m) = ∅, because of the claim. If max|j − `|, |k − m| < κ0, then |θ1

jk −θ1`m| ≥ 2R0L. By (2.4), tz1

[SR0

(p1jk)]∩ tz1

[SR0

(p1`m)]

= ∅, and again SR0(p1jk) ∩

SR0(p1`m) = ∅.

Induction step: If ∃θijk ∈ [−r0, r0] s.t. SR0(pijk) : (j, k) ∈ I, i = 1, . . . , n are

pairwise disjoint, then ∃θijk ∈ [−r0, r0] s.t. SR0(pijk) : (j, k) ∈ I, i = 1, . . . , n + 1

are pairwise disjoint.

Fix (j, k) ∈ I. We divide pi`,m : 1 ≤ i ≤ n, (`,m) ∈ I into two groups:

(1) “Dangerous” (for pn+1jk ): ∃θ ∈ [−r0, r0] s.t. SR0

(pi`m) ∩ SR0(ϕθ(zn+1

jk )) 6= ∅;


(2) “Safe” (for pn+1jk ): not dangerous.

No matter how we define θn+1jk , SR0

(pn+1jk ) ∩ SR0

(pi`m) = ∅ for all safe pi`m. But

the dangerous pi`m will introduce constraints on the possible values of θn+1jk .

By the claim, if pi`1,m1, pi`2,m2

are dangerous for pn+1jk , then |`1− `2|, |m1−m2| <

κ0. It follows that there are at most 4κ20N dangerous points for a given pn+1

jk .

Let Wn+1(pi`m) := tzn+1

[SR0

(pi`m)]. Since Lip(tzn+1

) ≤ L, Wn+1(pi`m) is a closedinterval of length less than w0 := 2LR0.

If pi`m is dangerous for pn+1jk , then we call Wn+1(pi`m) a “dangerous interval” for

pn+1jk . Let Wn+1

jk denote the union of all dangerous intervals for pn+1jk , and define

Tn+1(j, k) :=⋃

|j′−j|,|k′−k|<κ0

Wn+1j′k′ .

This is a union of no more than 16κ40N intervals of length less than w0 each.

Cut [−r0, r0] into four equal “quarters”: Q1 := [−r0,− r02 ], . . . , Q4 := [ r02 , r0]. Ifwe subtract n < L/w intervals of length less than w from an interval of length L,then the remainder must contain at least one interval of length (L− nw)/(n+ 1).It follows that for every s = 1, . . . , 4,

Qs \ Tn+1(κ0b jκ0c, κ0b kκ0

c) ⊃ an interval of lengthr0

32κ40N + 2

− w0 10κ20w0.

Let τn+1(j, k) denote the

center of such an interval in Q1, when (b jκ0c, b kκ0

c) = (0, 0) mod 2,


c) = (1, 0) mod 2,


c) = (0, 1) mod 2,


c) = (1, 1) mod 2.

Define θn+1jk := τn+1

(κ0b jκ0

c, κ0b kκ0c)

+ 3w0σjk.

This belongs to [−r0, r0], because τn+1(κ0b jκ0

c, κ0b kκ0c)

is the center of an inter-

val in Qs of radius at least 5κ20w0 > 3w0σjk, and Qs ⊂ [−r0, r0]. Moreover, since

Lip(tzn+1) ≤ L and w0 = 2LR0, we have tzn+1

(SR0(pn+1jk )) ⊂

[θn+1jk −w0, θ

n+1jk +w0

],

which by the definition of τn+1(j, k), lies outside Tn+1(κ0b jκ0c, κ0b kκ0

c). So

tzn+1(SR0

(pn+1jk )) ⊂ [−r0, r0] \ Tn+1(κ0b jκ0

c, κ0b kκ0c). (2.5)

We use this to show that SR0(pn+1jk )∩SR0

(pi`m) = ∅ for (`,m) ∈ I, i ≤ n. If pi`mis safe for pn+1

jk , then there is nothing to prove. If it is dangerous, tzn+1

[SR0(pi`m)

]⊂

Wn+1jk ⊂ Tn+1

(κ0b jκ0

c, κ0b kκ0c). By (2.5), SR0

(pn+1jk ) ∩ SR0

(pi`m) = ∅.

Next we show that SR0(pn+1jk ) is disjoint from every SR0

(pn+1`m ) s.t. (`,m) 6= (j, k).

There are three cases:

1. max|j − `|, |k −m| ≥ κ0: Use the claim.

2. 0 < max|j − `|, |k − m| < κ0 and (b jκ0c, b kκ0

c) = (b `κ0c, bmκ0

c): In this case

|θn+1jk − θn+1

`m | ≥ 3w0. Since tzn+1

[SR0

(pn+1jk )

]⊂ [θn+1

jk − w0, θn+1jk + w0] and

tzn+1

[SR0

(pn+1`m )

]⊂ [θn+1

`m −w0, θn+1`m +w0], tzn+1

[SR0

(pn+1jk )

]∩tzn+1

[SR0

(pn+1`m )

]=

∅. So SR0(pn+1jk ) ∩ SR0(pn+1

`m ) = ∅.


3. 0 < max|j − `|, |k −m| < κ0 and (b jκ0c, b kκ0

c) 6= (b `κ0c, bmκ0

c): In this case

max∣∣b jκ0

c − b `κ0c∣∣, ∣∣b kκ0

c − bmκ0c∣∣ = 1,

so τn+1(κ0b jκ0

c, κ0b kκ0c), τn+1

(κ0b `κ0

c, κ0bmκ0c)

fall in different Qs. Necessarily

tzn+1

[SR0

(pn+1jk )

]∩ tzn+1

[SR0

(pn+1`m )

]= ∅, so SR0

(pn+1jk ) ∩ SR0

(pn+1`m ) = ∅.

This concludes the inductive step, and the construction of θijk.

Completion of the proof: For every r ∈ [ρ0, R0], Λr :=⊎Ni=1

⊎(j,k)∈I Sr(p

ijk)

is a standard Poincare section with roof function bounded above by h0.

We saw that the union is disjoint for r = R0, therefore it’s disjoint for all r ≤ R0.We’ll show that the union is a Poincare section with roof function bounded by h0

for r = ρ0, and then this statement will follow for all r ≥ ρ0.Given p ∈ M , we must find 0 < R < h0 s.t. ϕR(p) ∈ Λρ0

. Since M ⊂⋃Ni=1 FBr0(zi), ∃i s.t. ϕ4r0(p) ∈ FBr0(zi). Therefore ϕ4r0(p) = ϕt(z) for some

z ∈ Sr0(zi), |t| < r0, whence ϕ4r0−t(p) ∈ Sr0(zi).Write ϕ4r0−t(p) = Ji(x, y) for some (x, y) s.t. x2 +y2 ≤ r2

0, and choose (j, k) ∈ Is.t. |x−jδ0|, |y−kδ0| < δ0. Since Ji is

√2–bi-Lipschitz, distM (ϕ4r0−t(p), zijk) < 2δ0.

It follows that distM (ϕ4r0−t(p), pijk) < 2δ0 + r0S0 < d. This places ϕ4r0−t(p)

inside FBrf (pijk). Let distS denote the intrinsic distance on Srs(pijk). We have

distS ≤ 2 distM (see Lemma 2.1), therefore, since pijk = qpijk(zijk),

distS(qpijk(ϕ4r0−t(p)), pijk) = distS(qpijk(ϕ4r0−t(p)), qpijk(zijk)) ≤

≤ 2 distM (qpijk(ϕ4r0−t(p)), qpijk(zijk)) ≤ 2LdistM (ϕ4r0−t(p), zijk) < 4Lδ0 < ρ0.

So ϕR(p) ∈ Sρ0(pijk) ⊂ Λρ0 for R := 4r0 − t− tpijk(ϕ4r0−t(p)).

Now |tpijk(ϕ4r0−t(p))| ≤ 2r0, because |θijk| ≤ r0 and |tpijk(ϕ4r0−t(p)) + θijk| =

|tpijk(ϕ4r0−t(p)) − tpijk(zijk)| ≤ L distM (ϕ4r0−t(p), zijk) < 2Lδ0 < r0. Also |t| < r0.

So r0 < R < 7r0. Since r0 <19h0, 0 < R < h0.

Theorem 2.8. Every flow invariant probability measure µ has adapted standardPoincare sections with arbitrarily small roof functions.

Proof. We use parameter selection, as in [LS82]. Let Λr := Λ(p1, . . . , pN ; r) (a ≤r ≤ b) be a one-parameter family of standard Poincare sections as in Lemma 2.7.We will show that Λr is adapted to µ for a.e. r.

Without loss of generality a, b, r, supRΛr ≤ h0 < r = 110

[min1,rs,rf ,dS0

], for all

r ∈ [a, b], where rs, rf , d are given by Lemmas 2.1–2.3, and S0 := 1 + max ‖Xp‖.We define the boundary of a canonical transverse disc Sr(p) by the formula

∂Sr(p) := expp(~v) : ~v ∈ TpM,~v ⊥ Xp, ‖~v‖p = r. Let

Sr :=⋃

qpi(∂Sr(pj)

): 1 ≤ i, j ≤ N ,distM (Sr(pi), Sr(pj)) ≤ h0S0

,

where qpi : FBrf (pi)→ Srf (pi) is given by Lemma 2.2.The assumption distM (Sr(pi), Sr(pj)) ≤ h0S0 ensures that ∂Sr(pj) ⊂ FBrf (pi),

because for every q ∈ ∂Sr(pj), distM (q, pi) ≤ diam[Sr(pj)]+distM (Sr(pj), Sr(pi))+diam[Sr(pi)] < h0S0 + 4r < 5rS0 < d, whence q ∈ Bd(pi) ⊂ FBrf (pi).

Claim. Sr contains the singular set of Λr.


Proof. Fix r and let R = RΛr , f = fΛr . We show that if p ∈ Λr \Sr, then f, f−1

are local diffeomorphisms on a neighborhood of p.Let i, j be the unique indices s.t. p ∈ Sr(pi) and f(p) ∈ Sr(pj). The speed of the

flow is less than S0, so distM (p, pj) ≤ distM (p, f(p))+distM (f(p), pj) < h0S0 +r <d. Thus p ∈ FBrf (pj). Similarly, distM (f(p), pi) < d, so f(p) ∈ FBrf (pi).

It follows that R(p) = −tpj (p) = |tpj (p)| and f(p) = qpj (p). Similarly, p =

qpi(f(p)). Since distM (Sr(pi), Sr(pj)) ≤ distM (p, ϕR(p)(p)) < h0S0 and p 6∈ Sr,

p 6∈ ∂Sr(pi) and f(p) 6∈ ∂Sr(pj).

So ∃V ⊂ Λr \ ∂Λr relatively open s.t. V 3 p and qpj (V ) ⊂ Λr \ ∂Λr. The mapqpj : V → qpj (V ) is a diffeomorphism, because qpj is differentiable and qpi qpj = Idon V . We will show that f W= qpj W on some open W ⊂ Λr \ ∂Λr containing p.R(p) = |tpj (p)|, so the curve ϕt(p) : 0 < t < |tpj (p)| does not intersect Λr. Λr

is compact and ϕ and tpj are continuous, so p has a relatively open neighborhoodW ⊂ V s.t. ϕt(q) : 0 < t < |tpj (q)| does not intersect Λr for all q ∈ W . Sof W= qpj W , and we see that f is a local diffeomorphism at p.

Similarly, f−1 is a local diffeomorphism at p, which proves that p 6∈ S(Λr).

The claim is proved, and we proceed to the proof of the theorem. We begin withsome reductions. Let fr := fΛr . By the claim it is enough to show that

µΛr

p ∈ Λr : lim inf

|n|→∞

1

|n|log distΛr (f

nr (p),Sr) < 0

= 0 for a.e. r ∈ (a, b). (2.6)

Indeed, this implies ∃r s.t. lim inf|n|→∞

1|n| log distΛr (f

nr (p),S(Λr)) ≥ 0 for µΛr–a.e.

p ∈ Λr, and the limit is non-positive, because distΛr (q,S(Λr)) ≤ distΛr (q, ∂Λ) ≤ rfor all q ∈ Λ. Let

Aα(r) := p ∈ Λb : ∃ infinitely many n ∈ Z s.t. 1|n| log distΛb(f

nb (p),Sr) < −α.

Λr ⊂ Λb, so µΛr µΛb , distΛr ≥ distΛb , and fr(x) = fn(x)b (x) with 1 ≤ n(x) ≤

supRrinf Rb

. Therefore (2.6) follows from the statement

∀ α > 0 rational(µΛb [Aα(r)] = 0 for a.e. r ∈ [a, b]

). (2.7)

Let Iα(p) := a ≤ r ≤ b : p ∈ Aα(r), then 1Aα(r)(p) = 1Iα(p)(r), whence by

Fubini’s Theorem∫ baµΛb [Aα(r)]dr =

∫Λb

Leb[Iα(p)]dµΛb(p). So (2.7) follows from

Leb[Iα(p)] = 0 for all p ∈ Λb. (2.8)

In summary, (2.8) ⇒ (2.7) ⇒ (2.6) ⇒ the theorem.

Proof of (2.8): Fix p ∈ Λb. If r ∈ Iα(p), then distΛb(fnb (p),Sr) < e−α|n| for

infinitely many n ∈ Z. Λb is a finite union of canonical transverse discs Sb(pi), andSb(pi) ∩ Sr is a finite union of projections Sb(pi) ∩ qpi(∂Sr(pj)), each satisfyingdistM (Sr(pi), Sr(pj)) ≤ h0S0. It follows that there are infinitely many n ∈ Z suchthat for some fixed 1 ≤ i, j ≤ N s.t. distM (Sr(pi), Sr(pj)) ≤ h0S0, it holds that

fnb (p) ∈ Sb(pi) and distΛb(fnb (p), qpi(∂Sr(pj))) < e−α|n|. (2.9)

Since distM (Sr(pi), Sr(pj)) ≤ h0S0, distM (pi, pj) ≤ h0S0 + 2r < d. This, and ourassumptions on b and h0 guarantee that Sb(pj), f

nb (p), and qpi(∂Sr(pj)) are inside

FBrf (pi) ∩ FBrf (pj) = domain of definition of qpi and qpj .


Suppose n satisfies (2.9). Let q be the point which minimizes distΛb(fnb (p), qpi(q))

over all q ∈ ∂Sr(pj), then

e−α|n| > distΛb(fnb (p), qpi(q))

≥ distM (fnb (p), qpi(q))

≥ L−1 distM (qpj [fnb (p)], qpj [qpi(q)]) ∵ Lip(qj) ≤ L

= L−1 distM (qpj [fnb (p)], q) ∵ q ∈ Sr(pj)

≥ (2L)−1 distΛb(qpj [fnb (p)], q). ∵ distSb(pj) ≤ 2 distM

So distΛb(qpj [fnb (p)], q) ≤ 2Le−α|n|. It follows that

|distΛb(pj , qpj [fnb (p)])− distΛb(pj , q)| ≤ 2Le−α|n|.

We now use the special geometry of canonical transverse discs: q ∈ ∂Sr(pj), sodistΛb(pj , q) = r. Writing Djn(p) := distΛb(pj , qpj [f

nb (p)]), we see that for every n

which satisfies (2.9), |Djn(p)− r| ≤ 2Le−α|n|. Thus every r ∈ Iα(p) belongs to

N⋃j=1

r ∈ [a, b] : ∃ infinitely many n ∈ Z s.t. |r −Djn(p)| ≤ 2Le−α|n|.

By the Borel-Cantelli Lemma, this set has zero Lebesgue measure.

Two standard Poincare sections with the same set of centers are called concentric.Since b/a in the last proof can be chosen arbitrarily large, that proof shows

Corollary 2.9. Let µ be a flow invariant probability measure. For every h0 > 0there are two concentric standard Poincare sections Λi = Λ(p1, . . . , pN ; ri) withheight functions bounded above by h0, s.t. Λ1 is adapted to µ and r2 > 2r1.

To see this take r1 close to a s.t. Λr1 is adapted, and r2 = b.

3. Pesin charts for adapted Poincare sections

One of the central tools in Pesin theory is a system of local coordinates whichpresent a non-uniformly hyperbolic map as a perturbation of a uniformly hyper-bolic linear map [Pes76, KH95, BP07]. We will construct such coordinates for thePoincare map of an adapted Poincare section. Adaptability is used, as in [KSLP86],to control the size of the coordinate patches along typical orbits (Lemma 3.3).

Suppose µ is a ϕ–invariant probability measure on M , and assume that µ isχ0–hyperbolic for some χ0 > 0. We do not assume ergodicity.

Fix once and for all a standard Poincare section Λ = Λ(p1, . . . , pN ; r) for ϕ,which is adapted to µ. Set f := fΛ, R := RΛ,S := S(Λ), and let µΛ be theinduced measure on Λ.

Without loss of generality, there is a larger concentric standard Poincare section

Λ := Λ(p1, . . . , pN ; r) s.t. r > 2r. Λ ⊃ Λ, and distΛ(Λ, ∂Λ) > r. We’ll use Λ as asafety margin in the following definition of the exponential map of Λ:

Expx : ~v ∈ TxΛ : ‖~v‖x < r → Λ, Expx(~v) := γx(‖~v‖x),

where γx(·) is the geodesic in Λ s.t. γ(0) = x and γ(0) = ~v. This makes sense

even near ∂Λ, because every Λ–geodesic can be prolonged r units of distance into Λ

without falling off the edge. Notice that Λ–geodesics are usually not M–geodesics,therefore Expx is usually different from expx.


As in [Spi79, chapter 9], there are ρdom, ρim ∈ (0, r) s.t. for every x ∈ Λ, Expxis a bi-Lipschitz diffeomorphism from ~v ∈ TxΛ : ‖~v‖x <

√2ρdom onto a relative

neighborhood of y ∈ Λ : distΛ(y, x) < ρim, with bi-Lipschitz constant less than 2.

Non-uniform hyperbolicity. Since Λ is adapted to µ, µΛ(S) = 0. By Lemma2.6, for µΛ–a.e. x ∈ Λ, f has one Lyapunov exponent in (−∞,−χ0 inf R) and oneLyapunov exponent in (χ0 inf R,∞). Let χ := χ0 inf R.

Non-uniformly hyperbolic set: Let NUHχ(f) be the set of x ∈ Λ\⋃n∈Z f

−n(S)s.t. Tfn(x)Λ = Eu(fn(x))⊕ Es(fn(x)) (n ∈ Z), where Eu, Es are one-dimensionallinear subspaces, and

(i) limn→±∞

1n log ‖dfnx ~v‖ < −χ for all non-zero ~v ∈ Es(x),

(ii) limn→±∞

1n log ‖df−nx ~v‖ < −χ for all non-zero ~v ∈ Eu(x),

(iii) limn→±∞

1n log | sin](Es(fn(x)), Eu(fn(x)))| = 0,

(iv) dfxEs(x) = Es(f(x)) and dfxE

u(x) = Eu(f(x)).

By the Oseledets Theorem and Lemma 2.6, µΛ[NUHχ(f)] = 1.

Pesin charts. This is a system of coordinates on NUHχ(f) which simplifies theform of f = fΛ. The following definition is slightly different than in Pesin’s originalwork [Pes76], but the proofs are essentially the same.

Fix a measurable family of unit vectors eu(x) ∈ Eu(x), es(x) ∈ Es(x) on NUHχ(f).

Since dimEu/s(x) = 1, eu/s(x) are determined up to a sign. To make the choice,let (e1

x, e2x) be a continuous choice of basis for TxΛ so that 〈e1

x, e2x, Xx〉 has positive

orientation. Pick eu(x), es(x) s.t. ](e1x, e

s(x)) ∈ [0, π), and ](es(x), eu(x)) > 0.

Pesin parameters: Given x ∈ NUHχ(f), let

α(x) := ](es(x), eu(x)),

s(x) :=√

2(∑∞

k=0 e2kχ‖dfkx es(x)‖2fk(x)

) 12

,

u(x) :=√

2(∑∞

k=0 e2kχ‖df−kx eu(x)‖2f−k(x)

) 12

.

The infinite series converge, because x ∈ NUHχ(f).

Oseledets-Pesin Reduction: Define a linear transformation Cχ(x) : R2 → TxΛ

by mapping(

10

)7→ s(x)−1es(x) and

(01

)7→ u(x)−1eu(x).

This diagonalizes the derivative cocycle dfx : TxM → Tf(x)M :

Theorem 3.1. ∃Cϕ s.t. ∀x ∈ NUHχ(f), Cχ(f(x))−1 dfx Cχ(x) =(Ax 00 Bx

),

where C−1ϕ ≤ |Ax| ≤ e−χ, and eχ ≤ |Bx| ≤ Cϕ.

The proof is a routine modification of the proofs in [BP07, theorem 3.5.5] or [KH95,theorem S.2.10], using the uniform bounds on df Λ\S (Lemma 2.5).

Our conventions for es(x), eu(x) guarantee that Cχ(x) is orientation-preserving,and one can show exactly as in [Sar13, Lemmas 2.4, 2.5] that

‖Cχ(x)‖ ≤ 1 and1√2

√s(x)2 + u(x)2

| sinα(x)|≤ ‖Cχ(x)−1‖ ≤

√s(x)2 + u(x)2

| sinα(x)|. (3.1)

We see that ‖Cχ(x)−1‖ is large exactly when Es(x) ≈ Eu(x) (small | sinα(x)|), orwhen it takes a long time to notice the exponential decay of 1

n log ‖dfnx es(x)‖ or of1n log ‖df−nx eu(x)‖ (large s(x) or u(x)). Large ‖Cχ(x)−1‖ means bad hyperbolicity.


Pesin Maps: The Pesin map at x ∈ NUHχ(f) (not to be confused with the Pesin

chart defined below) is Ψx : [−ρdom, ρdom]2 → Λ, given by

Ψx(u, v) = Expx

[Cχ(x)

(u

v

)].

Ψx is orientation-preserving, and it maps [−ρdom, ρdom]2 diffeomorphically onto a

neighborhood of x in Λ \ ∂Λ. Lip(Ψx) ≤ 2, because ‖Cχ(x)‖ ≤ 1. But Lip(Ψ−1x ) is

not uniformly bounded, because ‖Cχ(x)−1‖ can be arbitrarily large.

Maximal size: Fix some parameter 0 < ε < 2−32 (which will be calibrated later).

Although Ψx is well-defined on all of [−ρdom, ρdom]2, it will only be useful for us onthe smaller set [−Qε(x), Qε(x)]2, where

Qε(x) :=

ε3/β

(√s(x)2 + u(x)2

| sinα(x)|

)−12/β

∧(εdistΛ(x,S)

)∧ ρdom

ε

. (3.2)

Here S is the singular set of Λ, a ∧ b := mina, b, β is the constant in the C1+β

assumption on ϕ, and btcε := maxθ ∈ Iε : θ ≤ t where Iε := e− 13 `ε : ` ∈ N.

Qε(x) is called the maximal size (of the Pesin charts defined below).2 Noticethat Qε ≤ ε3/β‖C−1

χ ‖−(large power), so Qε(x) is small when x is close to S or whenthe hyperbolicity at x is bad. Another important property of Qε is that thanks tothe inequalities ‖Cχ‖ ≤ 1 and Qε < 2−

32 distΛ(x,S),

Ψx

([−Qε(x), Qε(x)]2

)⊂ Λ \S. (3.3)

This is in contrast to Ψx

([−ρdom, ρdom]2

), which may intersect S or Λ \ Λ.

Pesin Charts: The maximal Pesin chart at x ∈ NUHχ(f) (with parameter ε) isΨx : [−Qε(x), Qε(x)]2 → Λ \ S, Ψx(u, v) = Expx[Cχ(x)

(uv

)]. The Pesin chart of

size η is Ψηx := Ψx [−η,η]2 for 0 < η ≤ Qε(x).

The Pesin charts provide a system of local coordinates on a neighborhood ofNUHχ(f). The following theorem says that the Poincare map “in coordinates”

fx := Ψ−1f(x) f Ψx : [−Qε(x), Qε(x)]2 → R2

is close to a uniformly hyperbolic linear map.In what follows, 0 =

(00

)and “for all ε small enough P holds” means “∃ε0 > 0

which depends only on M,ϕ,Λ, β and χ0 s.t. for all ε < ε0, P holds”.

Theorem 3.2 (Pesin). For all ε small enough, for every x ∈ NUHχ(f), fx iswell-defined and injective on [−Qε(x), Qε(x)]2, and can be put there in the formfx(u, v) =

(Axu+ h1

x(u, v), Bxv + h2x(u, v)

), where

(1) C−1ϕ ≤ |Ax| ≤ e−χ and eχ ≤ |Bx| ≤ Cϕ, with Cϕ as in Theorem 3.1;

(2) hix are C1+ β2 functions s.t. hix(0) = 0, (∇hix)(0) = 0;

(3) ‖hix‖C1+

β2< ε on [−Qε(x), Qε(x)]2.

A similar statement holds for f−1x := Ψf−1(x) f−1 Ψx : [−Qε(x), Qε(x)]2 → R2.

Proof. Let U := Ψx([−Qε(x), Qε(x)]2). By (3.3), f and f−1 are C1+β on U , withuniform bounds on their C1+β norms (Lemma 2.5). Now continue as in [Sar13,Theorem 2.7] or [BP07, Theorem 5.6.1], replacing M by Λ and expp by Expp.

2We do not claim that Theorem 3.2 below does not hold on larger boxes [−Q′, Q′]2.


Adaptability and temperedness. The maximal size of Pesin charts may not bebounded below on NUHχ(f). A central idea in Pesin theory is that it is neverthelesspossible to control how fast Qε decays along typical orbits.

Define for this purpose the set NUH∗χ(f) of all x ∈ NUHχ(f) which on top ofthe defining properties (i)–(iv) of NUHχ(f) also satisfy

(v) limn→±∞

1n log distΛ(fn(x),S) = 0,

(vi) limn→±∞

1n log ‖Cχ(fn(x))−1‖ = 0,

(vii) limn→±∞

1n log ‖Cχ(fn(x))v‖ = 0 for v =

(10

),(

01

),

(viii) limn→±∞

1n log |detCχ(fn(x))| = 0.

Lemma 3.3. NUH∗χ(f) is an f–invariant Borel set of full µΛ–measure, and for

every x ∈ NUH∗χ(f), limn→±∞

1n logQε(f

n(x)) = 0.

Proof. (v) holds µΛ–a.e. because Λ is adapted to µΛ. (vi)–(viii) hold a.e., becauseof the Oseledets Theorem (apply the proof of [Sar13, Lemma 2.6] to the ergodiccomponents of µΛ). By (3.1), (v),(vi) imply 1

n logQε(fn(x))→ 0.

Lemma 3.4 (Pesin’s Temperedness Lemma). There exists a positive Borel functionqε : NUH∗χ(f) → (0, 1) s.t. for every x ∈ NUH∗χ(f), 0 < qε(x) ≤ εQε(x) and

e−ε/3 ≤ qε f/qε ≤ eε/3.

This follows from Lemma 3.3 as in [BP07, Lemma 3.5.7]. By Lemma 3.4,

Qε(fn(x)) > e−

13 |n|εqε(x) for all n ∈ Z. (3.4)

This control on the decay of Qε on typical orbits will be crucial for us.

Overlapping Pesin charts. Theorem 3.2 says that fx := Ψ−1f(x) f Ψx is close

to a linear hyperbolic map. This property is stable under perturbations, thereforewe expect fxy := Ψ−1

y f Ψx to be close to a linear hyperbolic map, whenever Ψy

is “sufficiently close” to Ψf(x). We now specify the meaning of “sufficiently close”.Recall that Λ is the disjoint union of a finite number of canonical transverse discs

Sr(pi). Let Dr(pi) := Sr(pi) \ ∂Sr(pi) = exppi(~v) : ~v ⊥ Xpi , ‖~v‖ < r. Choose for

every D = Dr(pi) a map Θ : TD → R2 s.t.

(1) Θ : TxD → R2 is a linear isometry for all x ∈ D.(2) Let ϑx : (Θ TxD)−1 : R2 → TxD, then (x, u) 7→ (Expx ϑx)(u) is smooth

and Lipschitz on Λ × u ∈ R2 : ‖u‖ < ρdom with respect to the metricd((x, u), (x′, u′)

):= distΛ(x, x′) + ‖u− u′‖.

(3) x 7→ ϑ−1x Exp−1

x is a Lipschitz map from D to C2(D,R2), the space of C2

maps from D to R2.

Recall that the Pesin map is Ψx(u, v) = Expx[Cχ(x)(uv

)], and the Pesin chart of

size 0 < η < Qε(x) is Ψηx := Ψx [−η,η]2 .

Overlapping charts: Let x1, x2 ∈ NUHχ(f). We say that Ψη1x1,Ψη2

x2ε–overlap,

and write Ψη1x1

ε≈ Ψη2

x2, if x1, x2 lie in the same transversal disc of Λ, e−ε < η1

η2< eε,

and distΛ(x1, x2) + ‖Θ Cχ(x1)−Θ Cχ(x2)‖ < η41η

42 .

Proposition 3.5. The following holds for all ε small enough: If Ψη1x1

ε≈ Ψη2

x2, then

(1) Ψηixi chart nearly the same patch: Ψxi

([−e−2εηi, e

−2εηi]2)⊂ Ψxj

([−ηj , ηj ]2

),


(2) Ψxi define nearly the same coordinates: distC1+

β2

(Ψ−1xi Ψxj , Id) < εη2

i η2j , where

the C1+ β2 –distance is calculated on [−e−ερdom, e

−ερdom]2.

Corollary 3.6. For all ε small enough, if x, y ∈ NUHχ(f) and Ψη′

f(x)

ε≈ Ψη

y, then

fxy := Ψ−1y f Ψx : [−Qε(x), Qε(x)]2 → R2 is well-defined, injective, and can be

put in the form fxy(u, v) =(Axyu+ h1

xy(u, v), Bxyv + h2xy(u, v)

), where

(1) C−1ϕ ≤ |Axy| ≤ e−χ and eχ ≤ |Bxy| ≤ Cϕ, with Cϕ as in Theorem 3.1;

(2) |hixy(0)| < εη, ‖∇hixy(0)‖ < εηβ/3;

(3) ‖hixy‖C1+

β3< ε (i = 1, 2) on [−Qε(x), Qε(x)]2.

A similar statement holds for f−1xy , assuming Ψη′

f−1(y)

ε≈ Ψη

x.

The proofs are routine modifications of the proofs in [Sar13]: Replace M by oneof the canonical transverse discs in Λ, and replace expx by Expx.

Part 2. Symbolic dynamics

Throughout this part we assume that M,X and ϕ satisfy our standing assump-tions, and that µ is a χ0–hyperbolic ϕ–invariant probability measure on M . Wefix a standard Poincare section Λ = Λ(p1, . . . , pN ; r) adapted to µ, and a larger

concentric standard section Λ := Λ(p1, . . . , pN ; r) s.t. r > 2r. Let f,R and Sdenote the Poincare map, roof function, and singular set of Λ, and let χ := χ0 inf R(a bound for the Lyapunov exponents of f µΛ–a.e., see Lemma 2.6).

Suppose P is a property. “For all ε > 0 small enough P holds” means “∃ε0 > 0which only depends on M , ϕ, Λ, β, χ0 s.t. P holds for all 0 < ε < ε0.”

In this part of the paper we construct a countable Markov partition for f on aset of full measure with respect to µΛ, and we use it to develop symbolic dynamicsfor ϕ. This was done in [Sar13] for surface diffeomorphisms, and the proof wouldhave applied to our setup verbatim had S been empty. We will indicate the changesneeded to treat the case S 6= ∅.

Not many changes are needed, because most of the work is done inside Pesincharts, where f and f−1 are smooth with uniformly bounded C1+β norm. One pointis worth mentioning, though: [Sar13] uses a uniform bound on | lnQε(f(x))/Qε(x)|,whereQε(x) is the maximal size of a Pesin chart. This quantity is no longer boundedwhen S 6= ∅. When this or other effects of S matter, we will give complete details.Otherwise, we will just sketch the general idea and refer to [Sar13] for details.

4. Generalized pseudo-orbits and shadowing

Generalized pseudo-orbits (gpo). Fix some small ε > 0. Recall that a pseudo-orbit with parameter ε is a sequence of points xii∈Z satisfying the nearest neighborconditions dist(f(xi), xi+1) < ε for all i ∈ Z. A gpo is also a sequence of objectssatisfying nearest neighbor conditions, but the objects and the conditions are morecomplicated, because of the need to record the hyperbolic features of each point:

ε–Double charts: Ordered pairs Ψpu,ps

x := (Ψx [−pu,pu]2 ,Ψx [−ps,ps]2) wherex ∈ NUHχ(f) and 0 < pu, ps ≤ Qε(x) (same Pesin chart, different domains).

ε–Generalized pseudo-orbits (gpo): A sequence Ψpui ,psi

xi i∈Z of ε–double chartswhich satisfies the following nearest neighbor conditions for all i ∈ Z:


(GPO1) Ψpui+1∧p

si+1

f(xi)

ε≈ Ψ

pui+1∧psi+1

xi+1 and Ψpui ∧p

si

f−1(xi+1)

ε≈ Ψ

pui ∧psi

xi , cf. Prop. 3.5.

(GPO2) pui+1 = mineεpui , Qε(xi+1) and psi = mineεpsi+1, Qε(xi).

A positive gpo is a one-sided sequence Ψpui ,psi

xi i≥0 with (GPO1), (GPO2). A neg-

ative gpo is a one-sided sequence Ψpui ,psi

xi i≤0 with (GPO1), (GPO2). Gpos werecalled “chains” in [Sar13].

Shadowing: A gpo Ψpui ,psi

xi i∈Z shadows the orbit of x, if f i(x) ∈ Ψxi

([−ηi, ηi]2

)for all i ∈ Z, where ηi := pui ∧ psi .

This notation is heavy, so we will sometime abbreviate it by writing vi instead

of Ψpui ,p

si

xi , and letting pu(vi) := pui , ps(vi) := psi , x(vi) := xi. The nearest neighbor

conditions [(GPO1) + (GPO2)] will be expressed by the notation viε−→ vi+1.

Lemma 4.1. Suppose 0 < pui , psi ≤ Qi satisfy pui+1 = mineεpui , Qi+1 and psi =

mineεpsi+1, Qi for i = 0, 1. If ηi := pui ∧ psi , then ηi+1/ηi ∈ [e−ε, eε].

Proof. See [Sar13], Lemma 4.4.

Admissible manifolds: An s–admissible manifold in a ε–double chart v = Ψpu,ps

x

is a set of the form Ψx(t, F (t)) : |t| ≤ ps, where F : [−ps, ps]→ R satisfies:

(Ad1) |F (0)| ≤ 10−3(pu ∧ ps).(Ad2) |F ′(0)| ≤ 1

2 (pu ∧ ps)β/3.

(Ad3) F is C1+ β3 and sup |F ′|+ Holβ/3(F ) ≤ 1

2 .

Similarly, a u–admissible manifold in v is a set of the form Ψx(F (t), t) : |t| ≤ pu,where F : [−pu, pu]→ R satisfies (Ad1–3).F is called the representing function, and the collections of all s/u–admissible

manifolds in v are denoted by M s(v) and M u(v).The representing function satisfies ‖F‖∞ ≤ Qε(x), because pu, ps ≤ Qε(x),

|F (0)| ≤ 10−3(pu ∧ ps) and |F ′| ≤ 12 .3 As a result, s/u–admissible manifolds are

subsets of Ψx([−Qε(x), Qε(x)]2), a set where f is smooth, and where if ε is smallenough then f is a perturbation of a uniformly hyperbolic linear map in Pesincoordinates (Theorem 3.2). This implies the following:

Graph Transform Lemma: For all ε small enough, if viε−→ vi+1, then the

forward image of a u–admissible manifold V u ∈ M u(vi) contains a unique u–admissible manifold Fu

vivi+1[V u], called the (forward) graph transform of V u.

Sketch of proof (see [Sar13, Prop. 4.12], [KH95, Supplement], or [BP07] for details):Let fxixi+1 := Ψ−1

xi+1f Ψxi : [−Qε(xi), Qε(xi)]2 → R2. By (GPO1) and Corollary

3.6, fxixi+1is ε close in the C1+ β

3 norm on [−Qε(xi), Qε(xi)]2 to a linear map whichcontracts the x–coordinate by at least e−χ and expands the y–coordinate by at leasteχ. Direct calculations show that if ε is much smaller than χ and V u ∈ M u(vi),then f(V u) ⊃ Ψxi+1

(G(t), t) : t ∈ [a, b] where G satisfies (Ad1–3), and [a, b] ⊃[−eχ/2pui , eχ/2pui ]. By (GPO2), if ε < χ/2 then [−eχ/2pui , eχ/2pui ] ⊃ [−pui+1, p

ui+1],

so f(V u) restricts to a u–admissible manifold in vi+1.

There is also a (backward) graph transform F svi+1vi : M s(vi+1) → M s(vi), ob-

tained by applying f−1 to s–admissible manifolds in vi+1 and restricting the resultto an s–admissible manifold in vi.

3In fact |F ′(t)| ≤ |F ′(0)|+ 12|t|β/3 ≤ ε for t ∈ dom(F ), since |t| ≤ pu/s ≤ Qε(x) ≤ ε3/β .


Put a metric on M u(vi) and M s(vi+1) by measuring the sup-norm distancebetween the representing functions. Using the form of fxixi+1 in coordinates, onecan show by direct calculations that Fu

vivi+1: M u(vi) →M u(vi+1) and F s

vi+1vi :

M s(vi+1)→M s(vi) contract distances by at least e−χ/2 [Sar13, Prop. 4.14].Suppose v− = vii≤0 is a negative gpo, and pick arbitrary V u−n ∈ M u(v−n)

(n ≥ 0), then V u0,n := (Fuv−1v0

· · · Fuv−n+1v−n+2

Fuv−nv−n+1

)(V u−n) ∈ M u(v0).

Using the uniform contraction of Fv−i−1v−i , it is easy to see that V u0,nn≥1 is aCauchy sequence, and that its limit is independent of the choice of V u−n [Sar13,Prop. 4.15]. Thus the following definition is proper for all ε small enough:

The unstable manifold of a negative gpo v−:

V u[v−] := limn→∞

(Fuv−1v0

· · · Fuv−n+1v−n+2

Fuv−nv−n+1

)(V u−n).

for some (any) choice of V u−n ∈M u(v−n).

Working with positive gpos and backward graph transforms, we can also makethe following definition:

The stable manifold of a positive gpo v+:

V s[v+] := limn→∞

(F sv1v0 · · · F s

vn−1vn−2F s

vnvn−1)(V sn ).

for some (any) choice of V sn ∈M s(vn).

The following properties hold:

(1) Admissibility: V u[v−] ∈ M u(v0) and V s[v+] ∈ M s(v0). This is becauseM u(v0),M s(v0) are closed in the supremum norm.

(2) Invariance: f−1(V u[vii≤0]) ⊂ V u[vii≤−1], f(V s[vii≥0]) ⊂ V s[vii≥1].This is immediate from the definition.

(3) Hyperbolicity: if x, y ∈ V u[v−], then distΛ(f−n(x), f−n(y)) −−−−→n→∞

0, and if

x, y ∈ V s[v+], then distΛ(fn(x), fn(y)) −−−−→n→∞

0. The rates are exponential.

To prove part (3) notice first that by the invariance property, fn(V s[v+]) andf−n(V u[v−]) remain inside Pesin charts. Therefore fn V s[v+] and f−n V u[v−] canbe written in Pesin coordinates as compositions of n uniformly hyperbolic maps onR2. One can then use direct calculations as in the proof of Pesin’s Stable ManifoldTheorem to prove (3). See e.g. [Sar13, Prop. 6.3].

The Shadowing Lemma.

Theorem 4.2. The following holds for all ε small enough: Every gpo with param-eter ε shadows a unique orbit.

Sketch of proof. Let v = vii∈Z be a gpo, vi = Ψpui ,p

si

xi . We have to show that thereexists a unique x s.t. f i(x) ∈ Ψxi([−ηi, ηi]2) for all i ∈ Z, where ηi = pui ∧ psi .V u := V u[vii≤0] and V s := V s[vii≥0] are admissible manifolds in v0. Be-

cause of properties (Ad1–3), V u and V s intersect at a unique point x, and x belongsto Ψx0

([−10−2η0, 10−2η0]2) [Sar13, Prop. 4.11], see also [KH95, Cor. S.3.8.]. Bythe invariance property,

fn(x) ∈ Ψxn([−Qε(xn), Qε(xn)]2) for all n ∈ Z.

We will show that v shadows x, and x is the only such point.


Any y s.t. fn(y) ∈ Ψxn([−Qε(xn), Qε(xn)]2) for all n ∈ Z equals x. The mapfxnxn+1 := Ψ−1

xn+1fΨxn is uniformly hyperbolic on [−Qε(xn), Qε(xn)]2. If Ψ−1

x0(x)

and Ψ−1x0

(y) have different y–coordinates, then successive application of fxnxn+1will

expand the difference between the y–coordinates of Ψ−1xn (fn(x)), Ψ−1

xn (fn(y)) expo-

nentially as n → ∞. If Ψ−1x0

(x),Ψ−1x0

(y) have different x–coordinates, then succes-

sive application of f−1x−n−1,x−n will expand the difference between the x–coordinates

of Ψ−1xn (fn(x)),Ψ−1

xn (fn(y)) exponentially as n → −∞. But these differences are

bounded by 2Qε(xn) whence by a constant, so Ψ−1x0

(x) = Ψ−1x0

(y), whence x = y.Let yk denote the unique intersection point of V u[vii≤k] and V s[vii≥k], then

fn(yk), fn+k(x) ∈ Ψxn+k([−Qε(xn+k), Qε(xn+k)]2) for all n ∈ Z. By the previous

paragraph, yk = fk(x). Since yk is the intersection of a u–admissible manifold andan s–admissible manifold in vk, yk ∈ Ψxk([−ηk, ηk]2) where ηk := puk ∧psk. It followsthat fk(x) ∈ Ψxk([−ηk, ηk]2) for all k ∈ Z.

Thus v shadows the orbit of x, and x is unique with this property.

Which points are shadowed by gpos? To appreciate the difficulty of this ques-tion, let us try the naıve approach: Given x ∈ NUHχ(f), set xi := f i(x), and look

for pui , psi s.t. Ψpui ,p

si

xi i∈Z is a gpo. (GPO1) is automatic, but without additionalinformation on Qε(f

i(x)), it is not clear that there exist pui , psi satisfying (GPO2).

This is where the adaptedness of Λ is used: For a.e. x, lim|n|→∞

1n logQε(f

n(x)) = 0,

whence by (3.4) there exist qε(x) > 0 s.t. Qε(fn(x)) > e−

13 ε|n|qε(x) > e−ε|n|qε(x)

for all n ∈ Z. So the following suprema range over non-empty sets:

pui := supt > 0 : Qε(fi−n(x)) ≥ e−εnt for all n ≥ 0,

psi := supt > 0 : Qε(fi+n(x)) ≥ e−εnt for all n ≥ 0.

It is easy to see that pui , psi satisfy (GPO2). Ψpui ,p

si

fi(x)i∈Z is a gpo which shadows x.

If we want to use the previous construction to shadow a set of full measure oforbits, then we need uncountably many “letters” Ψpu,ps

x . The following propositionachieves this with a countable discrete collection. Recall the definition of NUH∗χ(f)from Lemma 3.3, and let

NUH#χ (f) := x ∈ NUH∗χ(f) : lim sup

n→∞qε(f

n(x)), lim supn→−∞

qε(fn(x)) 6= 0. (4.1)

NUH#χ (f) has full measure by Poincare’s recurrence theorem. It is f–invariant.

Proposition 4.3. The following holds for all ε small enough. There exists a count-able collection of ε–double charts A with the following properties:

(1) Discreteness: Let D(x) := distΛ

(x, f(x), f−1(x),S

), then for every t > 0

the set Ψpu,ps

x ∈ A : D(x), pu, ps > t is finite.

(2) Sufficiency: For every x ∈ NUH#χ (f) there is a gpo v ∈ A Z which shadows

x, and which satisfies pu(vn) ∧ ps(vn) ≥ e−ε/3qε(fn(x)) for all n ∈ Z.(3) Relevance: For every v ∈ A there is a gpo v ∈ A Z s.t. v0 = v and v shadows

a point in NUHχ(f).

Proof. The proof for diffeomorphisms in [Sar13, Prop. 3.5 and 4.5] does not extendto our case, because it uses a uniform bound F−1 ≤ Qε f/Qε ≤ F which does nothold in the presence of singularities. We bypass this difficulty as follows.


Let X := [Λ \⋃i=−1,0,1 f

i(S)]3× (0,∞)3×GL(2,R), together with the product

topology, and let Y ⊂ X denote the subset of (x,Q,C) ∈ X of the form

x = (x, f(x), f−1(x)), where x ∈ NUH∗χ(f),

Q = (Qε(x), Qε(f(x)), Qε(f−1(x))),

C = (Cχ(x), Cχ(f(x)), Cχ(f−1(x))).

Cut Y into the countable disjoint union Y =⊎

(k,`)∈N40Yk,` where N0 = N∪0 and

Yk,` :=

(x,Q,C) ∈ Y :

x ∈ NUH∗χ(f), e−(k+1) < Qε(x) ≤ e−ke−(ì+1) < distΛ(f i(x),S) ≤ e−ì (i = 0, 1,−1)

.

Precompactness Lemma. Yk,` are precompact in X.

Proof. Suppose (x,Q,C) ∈ Yk,`. By (3.1) and (3.2), ‖Cχ(x)−1‖ ≤ ε14 e

β(k+1)12 . One

can show as in [Sar13, page 403], that C−1 ≤ ‖Cχ(y)−1‖/‖Cχ(f(y))−1‖ ≤ C for ally ∈ NUHχ(f) for some global constant C. It follows that

‖Cχ(f i(x))−1‖ ≤ Cε 14 e

β(k+1)12 for i = 0, 1,−1.

Since Cχ(·) is a contraction, C ∈ Gk × Gk × Gk, where Gk is the compact set

A ∈ GL(2,R) : ‖A‖ ≤ 1, ‖A−1‖ ≤ Cε 14 e

β(k+1)12 .

Next we bound Q in a compact set. By (3.1), if (x,Q,C) ∈ Yk,`, then

Qε(fi(x))

Qε(x)≥(√

2‖Cχ(f i(x))−1‖‖Cχ(x)−1‖

)− 12β

∧(

distΛ(f i(x),S)

distΛ(x,S)

)≥ 2−

6βC−

12β ∧e−(ì+1),

whence Qε(fi(x)) ≥ 2−

6βC−

12β e−(ì+1)e−(k+1). By definition, Qε(f

i(x)) ≤ ρdom, soQ ∈ (Fk,`)

3 with Fk,` ⊂ R compact.

Finally, x ∈ E`0×f(E`0)×f−1(E`0), with E`0 := y ∈ Λ : distΛ(y,S) ≥ e−`0−1.E`0 is compact because Λ is compact, and f(E`0), f−1(E`0) are compact becausef E`0 , f

−1 E`0 are continuous.

In summary, Yk,` ⊂∏i=0,1,−1 f

i(E`0)× (Fk,`)3× (Gk)3, a compact subset of X.

So Yk,` is precompact in X, proving the lemma.

Since Yk,` is precompact in X, Yk,` contains a finite set Yk,`(m) s.t. for every(x,Q,C) ∈ Yk,` there exists some (y,Q′, C ′) ∈ Yk,`(m) s.t. for every |i| ≤ 1,

(a) distΛ(f i(x), f i(y)) + ‖Θ Cχ(f i(x))−Θ Cχ(f i(y))‖ < e−8(m+3),

(b) e−ε/3 < Qε(fi(x))/Qε(f

i(y)) < eε/3.

(Θ is defined at the end of §3.)

Definition of A : The set of double charts Ψpu,ps

x s.t. for some k, `0, `1, `−1,m ≥ 0

(A1) x is the first coordinate of some (x,Q,C) ∈ Yk,`(m);

(A2) 0 < pu, ps ≤ Qε(x) and pu, ps ∈ Iε = e− 13 `ε : ` ∈ N;

(A3) pu ∧ ps ∈ [e−m−2, e−m+2].

Proof that A is discrete: Fix t > 0. Suppose Ψpu,ps

x ∈ A and let k, `,m beas above. If D(x), pu, ps > t, then:

k ≤ | log t| because t < pu ≤ Qε(x) ≤ e−k; ì ≤ | log t| because t < D(x) ≤ distΛ(f i(x),S) ≤ e−ì ; m ≤ | log t|+ 2 because t < pu ∧ ps ≤ e−m+2.


So #x : Ψpu,ps

x ∈ A , D(x), pu, ps > t ≤

⌈| log t|

⌉∑k,`0,`1,`−1=0

⌈| log t|

⌉+2∑

m=0#Yk,`(m) < ∞.

Also, #(pu, ps) : Ψpu,ps

x ∈ A , D(x), pu, ps > t ≤ (#(Iε ∩ [t, 1]))2 < ∞. Thus#Ψpu,ps

x ∈ A : D(x), pu, ps > t <∞, proving that A is discrete.

The proof of sufficiency requires some preparation. A sequence (pun, psn)n∈Z iscalled ε–subordinated to a sequence Qnn∈Z ⊂ Iε, if 0 < pun, p

sn ≤ Qn; pun, p

sn ∈ Iε;

pun+1 = mineεpun, Qn+1 and psn−1 = mineεpsn, Qn−1 for all n.

First Subordination Lemma. Let qnn∈Z, Qnn∈Z ⊂ Iε. If for every n ∈ Z0 < qn ≤ Qn and e−ε ≤ qn/qn+1 ≤ eε, then there exists (pun, psn)n∈Z which isε–subordinated to Qnn∈Z, and such that pun ∧ pun ≥ qn for all n.

Second Subordination Lemma. Suppose (pun, psn)n∈Z is ε–subordinated toQnn∈Z. If lim sup

n→∞(pun ∧ psn) > 0 and lim sup

n→−∞(pun ∧ psn) > 0, then pun (resp. psn) is

equal to Qn for infinitely many n > 0, and for infinitely many n < 0.

These are Lemmas 4.6 and 4.7 in [Sar13].

Proof of sufficiency: Fix x ∈ NUH#χ (f). Recall the definition of qε(·) from

Pesin’s Temperedness Lemma (Lemma 3.4), and choose qn ∈ Iε s.t. qn/qε(fn(x)) ∈

[e−ε/3, eε/3]. Necessarily e−ε ≤ qn/qn+1 ≤ eε.By the first subordination lemma there exists (qun, qsn)n∈Z s.t. (qun, qsn)n∈Z

is ε–subordinated to e−ε/3Qε(fn(x))n∈Z, and qun ∧ qsn ≥ qn for all n ∈ Z. Letηn := qun∧qsn. By Lemma 4.1, e−ε ≤ ηn+1/ηn ≤ eε. Since ηn ≥ qn ≥ e−ε/3qε(fn(x))

and x ∈ NUH#χ (f), lim supn→±∞ ηn > 0.

Choose non-negative integers mn, kn, `n = (`n0 , `n1 , `

n−1) s.t. for all n ∈ Z,

ηn ∈ [e−mn−1, e−mn+1], Qε(fn(x)) ∈ (e−kn−1, e−kn ], distΛ(fn+i(x),S) ∈ (e−`

ni −1, e−`

ni ] for i = 0, 1,−1.

Choose an element of Ykn,`n with first coordinate fn(x), and approximate it bysome element of Ykn,`n(mn) with first coordinate xn s.t. for i = 0, 1,−1

(an) distΛ(f i(fn(x)), f i(xn))+‖ΘCχ(f i(fn(x)))−ΘCχ(f i(xn))‖ < e−8(mn+3),

(bn) e−ε/3 < Qε(fi(fn(x)))/Qε(f

i(xn)) < eε/3.

By (bn) with i = 0, Qε(xn) ≥ e−ε/3Qε(fn(x)) ≥ ηn. By the first subordination

lemma, there exists (pun, psn)n∈Z ε–subordinated to Qε(xn)n∈Z such that pun ∧psn ≥ ηn for all n ∈ Z. Necessarily, pun ∧ psn ≥ e−ε/3qε(fn(x)). Let

v := Ψpun,psn

xn n∈Z.

We will show that v ∈ A Z, v is a gpo, and v shadows the orbit of x.

Proof that Ψpun,p

sn

xn ∈ A : (A1), (A2) are clear, so we focus on (A3). It is enoughto show that 1 ≤ (pun ∧ psn)/ηn ≤ e. The lower bound is by construction. Forthe upper bound, recall that lim supn→±∞ ηn > 0, so by the second subordination

lemma qun = e−ε/3Qε(fn(x)) for infinitely many n < 0. By (bn) with i = 0, qun ≥


e−εQε(xn) ≥ e−εpun for infinitely many n < 0. If qun ≥ e−εpun, then qun+1 ≥ e−εpun+1:

qun+1 = mineεqun, e−ε/3Qε(fn+1(x))

≥ mineεe−εpun, e−2ε/3Qε(xn+1), by (bn+1) with i = 0

≥ e−ε mineεpun, Qε(xn+1)= e−εpun+1.

It follows that qun ≥ e−εpun for all n ∈ Z. Similarly qsn ≥ e−εpsn for all n ∈ Z, whenceηn ≥ e−ε(pun ∧ psn) for all n, giving us (A3).

Proof that Ψpun,psn

xn n∈Z is a gpo. (GPO2) is true by construction, so we just needto check (GPO1). We write (an) with i = 1, and (an+1) with i = 0:

distΛ(fn+1(x), f(xn)) + ‖Θ Cχ(fn+1(x))−Θ Cχ(f(xn))‖ < e−8(mn+3),

distΛ(fn+1(x), xn+1) + ‖Θ Cχ(fn+1(x))−Θ Cχ(xn+1)‖ < e−8(mn+1+3).

So xn+1, f(xn), fn+1(x) are all in the same canonical transverse disc, and

distΛ(f(xn), xn+1) + ‖Θ Cχ(f(xn))−Θ Cχ(xn+1)‖ < e−8(mn+3) + e−8(mn+1+3).(4.2)

The proof of (A3) above shows that ξn := pun ∧ psn ∈ [e−mn−2, e−mn+2]. Alsoξn/ξn+1 ∈ [e−ε, eε], because (pun, psn)n∈Z is ε–subordinated (see Lemma 4.1). Sothe right hand side of (4.2) is less than e−8(1 + e8ε)ξ8

n+1 < (pun+1 ∧ psn+1)8. Thus

Ψpun+1∧p

sn+1

f(xn)

ε≈ Ψ

pun+1∧psn+1

xn+1 . A similar argument with (an) and i = −1, and with

(an−1) and i = 0 shows that Ψpun−1∧p

sn−1

f−1(xn)

ε≈ Ψ

pun−1∧psn−1

xn−1 . So (GPO1) holds, and v isa gpo.

Proof that v shadows x: By (an) with i = 0, Ψpun∧p

sn

xn

ε≈ Ψ

pun∧psn

fn(x) for all n ∈ Z. By

Proposition 3.5, fn(x) = Ψfn(x)(0) ∈ Ψxn([−pun ∧ psn, pun ∧ psn]2). So v shadows x.

Arranging relevance: Call an element v ∈ A relevant, if there is a gpo v ∈ A Z

s.t. v0 = v and v shadows a point in NUHχ(f). In this case every vi is relevant,because NUHχ(f) is f–invariant. So A ′ := v ∈ A : v is relevant is sufficient. Itis discrete, because A ′ ⊆ A and A is discrete. The theorem follows with A ′.

The inverse shadowing problem. The same orbit can be shadowed by manydifferent gpos. The “inverse shadowing problem” is to control the set of gpos

Ψpui ,psi

xi i∈Z which shadow the orbit of a given point x. (GPO1) and (GPO2) weredesigned to make this possible. We need the following condition.

Regularity: Let A be as in Proposition 4.3. A gpo v ∈ A Z is called regular, ifvii≥0, vii≤0 have constant subsequences.

Proposition 4.4. Almost every x ∈ Λ is shadowed by a regular gpo in A Z.

Proof. We will show that this is the case for all x ∈ NUH#χ (f). Since A is sufficient

(Prop. 4.3(2)), for every x ∈ NUH#χ (f) there is a gpo v = Ψpuk ,p

sk

xk k∈Z ∈ A Z which

shadows x s.t. for all k ∈ Z, ηk := puk ∧ psk ≥ e−ε/3qε(fk(x)).

Since pu/s ≤ Qε(·) ≤ ε distΛ(·,S),

distΛ(xk,S) ≥ ε−1e−ε/3qε(fk(x)) for all k ∈ Z. (4.3)


Since vkε−→ vk+1, Ψ

ηk+1

f(xk)

ε≈ Ψ

ηk+1xk+1 , whence f(xk) ∈ Ψxk+1

([−Q(xk+1), Q(xk+1)]2).

Since Lip(Ψxk+1) ≤ 2, distΛ(f(xk), xk+1) ≤ 2

√2Q(xk+1) ≤ 3εdistΛ(xk+1,S). By

the triangle inequality, (4.3), and the inequality e−ε/3 ≤ qε f/qε ≤ eε/3,

distΛ(f(xk),S) ≥ distΛ(xk+1,S)− distΛ(f(xk), xk+1) ≥ (1− 3ε) distΛ(xk+1,S)

≥ (1− 3ε)ε−1e−ε/3qε(fk+1(x)) > (1− 3ε)ε−1e−εqε(f

k(x)) > qε(fk(x)),

provided ε is small enough. Similarly, distΛ(f−1(xk),S) > qε(fk(x)), and we obtain

that minD(xk), puk , psk ≥ e−ε/3qε(fk(x)) for all k ∈ Z.

Since x ∈ NUH#χ (f), ∃ki, ì ↑ ∞ and c > 0 s.t. qε(f

−ki(x)) ≥ c, qε(fì(x)) ≥ c.

Since A is discrete, there must be some constant subsequences v−kij , vìj .

The following proposition says in a precise way, that if u is a regular gpo whichshadows the orbit of x, then ui is determined “up to bounded error.” Togetherwith the discreteness of A , this implies that for every i there are only finitely manypossibilities for ui.

4

Theorem 4.5. The following holds for all ε small enough. Let u, v be regular gpos

which shadow the orbit of the same point x. If ui = Ψpui ,p

si

xi and vi = Ψqui ,q

si

yi , then

(1) distΛ(xi, yi) < 10−1 maxpui ∧ psi , qui ∧ qsi ;(2) distΛ(fk(xi),S)/distΛ(fk(yi),S) ∈ [e−

√ε, e√ε] for k = 0, 1,−1;

(3) Qε(xi)/Qε(yi) ∈ [e−3√ε, e

3√ε];

(4) (Ψ−1yi Ψxi) = (−1)σiId + ci + ∆i on [−ε, ε]2, where ‖ci‖ < 10−1(qui ∧ qsi ) and

σi ∈ 0, 1 are constants, and ∆i : [−ε, ε]2 → R2 is a vector field s.t. ∆i(0) = 0and ‖(d∆i)v‖ < 3

√ε on [−ε, ε]2;

(5) pui /qui , p

si/q

si ∈ [e−

3√ε, e

3√ε].

Proof. Denote the stable and unstable manifolds of u and v by Uu, Us and V u, V s.By the proof of the shadowing lemma, Uu ∩ Us = V u ∩ V s = x.

Part (1). Uu/s are admissible manifolds. By (Ad1–3), their intersection pointmust satisfy x = Ψx0

(ξ) where ‖ξ‖∞ ≤ 10−2(pu0 ∧ ps0), see [Sar13, Prop. 4.11].

Since Lip(Ψx0) ≤ 2, distΛ(x0, x) ≤ 1

50 (pu0 ∧ps0). Similarly, distΛ(y0, x) ≤ 150 (qu0 ∧qs0),

whence distΛ(x0, y0) ≤ 125 maxpu0 ∧ ps0, qu0 ∧ qs0.

Part (2). In what follows a = b± c means b− c ≤ a ≤ b+ c.

distΛ(x0,S) = distΛ(x,S)± distΛ(x, x0) = distΛ(x,S)± 50−1(pu0 ∧ ps0), by part 1

= distΛ(x,S)± 50−1Qε(x0), because pu0 , ps0 ≤ Qε(x0)

= distΛ(x,S)± 50−1εdistΛ(x0,S), by the definition of Qε(x0).

Therefore distΛ(x,S)distΛ(x0,S) = 1± 50−1ε. Similarly distΛ(x,S)

distΛ(y0,S) = 1± 50−1ε. It follows that

if ε is small enough then distΛ(x0,S)distΛ(y0,S) ∈ [e−ε, eε].

Applying this argument to suitable shifts of u, v, we obtain distΛ(x1,S)distΛ(y1,S) ∈ [e−ε, eε]

and distΛ(x−1,S)distΛ(y−1,S) ∈ [e−ε, eε].

4But the set of all possible full sequences u can be uncountable.


Since u0ε−→ u1, Ψ

pu1∧ps1

f(x0)

ε≈ Ψ

pu1∧ps1

x1 . So x1, f(x0) ∈ Ψx1

([−Qε(x1), Qε(x1)]2

).

Since Lip(Ψx1) ≤ 2, distΛ

(x1, f(x0)

)≤ 2√

2Qε(x1) < 6εdistΛ(x1,S). Thus

distΛ(f(x0),S) = distΛ(x1,S)± distΛ(x1, f(x0)) = distΛ(x1,S)± 6εdistΛ(x1,S)

= e±7ε distΛ(x1,S), provided ε is small enough.

Similarly, distΛ(f(y0),S) = e±7ε distΛ(y1,S). Since distΛ(x1,S)distΛ(y1,S) ∈ [e−ε, eε],

distΛ(f(x0),S)

distΛ(f(y0),S)∈ [e−15ε, e15ε] ⊂ [e−

√ε, e√ε], provided ε is small enough.

Similarly, one shows that distΛ(f−1(x0),S)distΛ(f−1(y0),S) ∈ [e−

√ε, e√ε].

Part (3). One shows as in [Sar13, §6 and §7] that for all ε small enough,

sinα(xi)

sinα(yi)∈ [e−

√ε, e√ε] and

s(xi)

s(yi),u(xi)

u(yi)∈ [e−4

√ε, e4

√ε]. (4.4)

The proof carries over without change, because all the calculations are done onfn(Us), fn(V s) (n ≥ 0) and fn(Uu), fn(V u) (n ≤ 0), and these sets stay insidePesin charts, away from S. By (4.4), for all ε small enough,(√

s(x0)2 + u(x0)2

| sinα(x0)|

)− 12β(√

s(y0)2 + u(y0)2

| sinα(y0)|

) 12β

∈ [e−3√ε, e−

3√ε].

By part (2) and the definition of Qε,Qε(x0)Qε(y0) ∈ [e−

3√ε, e

3√ε].

Part (4). This is done exactly as in [Sar13, §9], except that one needs to add theconstraint ε < ρdom to be able to use the smoothness of p 7→ Expp on Λ.

Part (5). This is done exactly as in the proof of [Sar13, Prop. 8.3], except thatstep 1 there should be replaced by part (3) here.

Remark. Regularity is needed in parts (3), (4), (5). Parts (3), (4) also use the fullforce of (Ad1–3), and Part (5) is based on (GPO2). See [Sar13].

5. Countable Markov partitions and symbolic dynamics

Sinai and Bowen gave several methods for constructing Markov partitions foruniformly hyperbolic diffeomorphisms. One of these, due to Bowen, uses pseudo-orbits and shadowing [Bow75]. The theory of gpos we developed in the previoussection allows us to apply this method to adapted Poincare sections. The result is aMarkov partition for f : Λ→ Λ. It is then a standard procedure to code f : Λ→ Λby a topological Markov shift, and ϕ : M →M by a topological Markov flow.

Step 1: A Markov extension. Let A be the countable set of double charts weconstructed in Proposition 4.3, and let G denote the countable directed graph with

set of vertices A and set of edges (u, v) ∈ A ×A : uε−→ v.

Lemma 5.1. Every vertex of G has finite ingoing degree, and finite outgoing degree.

Proof. We fix u ∈ A , and bound the number of v s.t. uε−→ v, using the discreteness

and relevance of A (cf. Prop. 4.3).

By the relevance property, uε−→ v extends to a path u

ε−→ vε−→ w. Write u =

Ψpu,ps

x , v = Ψqu,qs

y , w = Ψru,rs

z , then


ru∧rsqu∧qs ,

qu∧qspu∧ps ∈ [e−ε, eε], by Lemma 4.1.

distΛ(y,S) ≥ ε−1e−ε(pu ∧ ps), because distΛ(y,S) ≥ ε−1Qε(y), Qε(y) ≥ qu ∧ qs. distΛ(f(y),S) ≥ e−2ε(ε−1 − 3)(pu ∧ ps), because

distΛ(f(y),S) ≥ distΛ(z,S)− distΛ(z, f(y))

≥ ε−1Qε(z)− 2√

2Qε(z) ∵ f(y) ∈ Ψz([−Qε(z), Qε(z)]2) , Lip(Ψz) ≤ 2

≥ (ε−1 − 3)(ru ∧ rs) ≥ e−2ε(ε−1 − 3)(pu ∧ ps).

distΛ(f−1(y),S) ≥ e−2ε(ε−1 − 3)(pu ∧ ps), for similar reasons.

So D(y) := distΛ(y, f(y), f−1(y),S) ≥ t := e−2ε(ε−1 − 3)(pu ∧ ps).By the discreteness of A (and assuming ε < 1

3 ), #v ∈ A : uε−→ v < ∞. The

finiteness of the ingoing degree is proved in the same way.

The Markov Extension: Let Σ(G ) denote the set of two-sided paths on G :

Σ(G ) := v ∈ A Z : viε−→ vi+1 for all i ∈ Z.

We equip Σ(G ) with the metric d(u, v) = exp[−min|n| : un 6= vn], and with theaction of the left shift map σ : Σ(G ) → Σ(G ), σ : vii∈Z 7→ vi+1i∈Z. Σ(G ) isexactly the collection of gpos belonging to A Z, therefore π : Σ(G )→ Λ given by

π(v) := unique point whose f–orbit is shadowed by v

is well-defined. Necessarily f π = π σ, so σ : Σ(G ) → Σ(G ) is an extension off : Λ→ Λ (at least on a subset of full measure, by Prop. 4.4).

It is easy to see, using the finite degree of the vertices of G , that (Σ(G ), d) isa locally compact, complete and separable metric space. The left shift map is abi-Lipschitz homeomorphism. The subset of regular gpos

Σ#(G ) := v ∈ Σ(G ) : vii≤0, vii≥0 contain constant subsequenceshas full measure w.r.t. every σ–invariant Borel probability measure.

As we saw in the proof of the shadowing theorem (Thm. 4.2), π[v] is the uniqueintersection of V u(v−) and V s(v+) where v± are the half gpos determined by v.The proof shows that the following holds for all ε small enough:

(1) Holder continuity: π is Holder continuous (because Fu,F s are contrac-tions, see [Sar13, Thm 4.16(2)]).

(2) Almost surjectivity: µΛ(Λ \ π[Σ#(G )]) = 0 (Prop. 4.4).(3) Inverse Property: for all x ∈ Λ, i ∈ Z, #vi : v ∈ Σ#(G ), π(v) = x < ∞.

(Thm 4.5 and the discreteness of A .)

The inverse property does not imply that π is finite-to-one or even countable-to-one. The following steps will lead us to an a.e. finite-to-one Markov extension.

Step 2: A Markov cover. Given v ∈ A , let 0[v] := v ∈ Σ(G ) : v0 = v. This isa partition of Σ(G ). The projection to Λ

Z := Z(v) : v ∈ A , where Z(v) := π(v) : v ∈ Σ#(G ), v0 = vis not a partition. It could even be the case that Z(v) = Z(u) for u 6= v (in thiscase, we agree to think of Z(v), Z(u) as different elements of Z ). Here are someimportant properties of Z .

Covering property: Z covers a set of full µΛ–measure.

Proof. Z covers NUH#χ (f).


Local finiteness: For all Z ∈ Z , #Z ′ ∈ Z : Z ′ ∩ Z 6= ∅ < ∞. Even better:#u ∈ A : Z(u) ∩ Z 6= ∅ <∞ for all Z ∈ Z .

Proof. Write Z = Z(Ψpu,ps

x ). If Z(Ψqu,qs

y ) ∩ Z 6= ∅, then qu ∧ qs ≥ e−3√ε(pu ∧ ps)

and distΛ(y, f(y), f−1(y),S) ≥ e−√ε distΛ(x, f(x), f−1(x),S) (Theorem 4.5).

Since A is discrete, there are only finitely many such Ψqu,qs

y in A .

Product structure: Suppose v ∈ A and Z = Z(v). For every x ∈ Z there aresets Wu(x, Z) and W s(x, Z) called the s–fibre and u–fibre of x in Z s.t.

(1) Z =⋃x∈ZW

u(x, Z), Z =⋃x∈ZW

s(x, Z).(2) Any two s–fibres in Z are either equal or disjoint. Any two u–fibres in Z are

either equal or disjoint.(3) For every x, y ∈ Z, Wu(x, Z) ∩W s(y, Z) intersect at a unique point.

Notation: [x, y]Z := unique point in Wu(x, Z) ∩W s(x, Z) (“Smale bracket”).

Proof. Recall the notation for the stable and unstable manifolds of positive andnegative gpos (§4). Fix Z = Z(v) in Z , x ∈ Z, and let

V s(x, Z) := V s[vii≥0], for some (any) v ∈ Σ#(G ) s.t. v0 = v and π(v) = x. V u(x, Z) := V u[vii≤0], for some (any) v ∈ Σ#(G ) s.t. v0 = v and π(v) = x. W s(x, Z) := V s(x, Z) ∩ Z. Wu(x, Z) := V s(x, Z) ∩ Z.To see that the definition is proper, suppose u, v are two regular gpos such thatu0 = v0 = v. If V t(u), V t(v) intersect at some point z for t = s or u, thenV t(u) = V t(v), because both are equal to the piece of the local stable/unstablemanifold of z at Ψx(v)([−pt(v), pt(v)]2). See [Sar13, Prop. 6.4] for details. Inparticular, π(u) = π(v)⇒ V t[u] = V t[v] (t = u, s).

This argument also shows that any two t–fibres (t = s or u) are equal or disjoint.(1) is because Wu(x, Z), W s(x, Z) contain x. For (3), Write Wu(x, Z) = V u(u)∩Zand W s(y, Z) = V s(v) ∩ Z where u, v ∈ Σ#(G ) satisfy u0 = v0 = v. Let w :=(. . . , u−2, u−1, v, v1, v2, . . .) with the dot indicating the zeroth coordinate. Clearlyπ(w) ∈ Wu(x, Z) ∩W s(y, Z). Since Wu(x, Z) ∩W s(y, Z) ⊂ V u(x, Z) ∩ V s(y, Z)and a u–admissible manifold intersects an s–admissible at most once [KH95, Cor.S.3.8], [Sar13, Prop. 4.11], Wu(x, Z) ∩W s(y, Z) = π(w).Symbolic Markov property: If x = π(v) and v ∈ Σ#(G ), then

f[W s(x, Z(v0))

]⊂W s(f(x), Z(v1)) and f−1

[Wu(f(x), Z(v1))

]⊂Wu(x, Z(v0)).

Proof. Fix y ∈W s(x, Z(v0)). Choose u ∈ Σ#(G ) s.t. u0 = v0 and y = π(u). Write

ui = Ψqui ,q

si

yi and ηi := qui ∧ qsi , then f−n(y) ∈ Ψy−n([−η−n, η−n]2) for all n ≥ 0.

Write vi = Ψpui ,p

si

xi and ξi := pui ∧ psi . Since y ∈W s(x, Z(v0)) ⊂ V s(v+), fn(y) ∈fn[V s(v+)] ⊂ V s(σnv+) ⊂ Ψxn([−ξn, ξn]2) for all n ≥ 0.

It follows that the gpo w = (. . . , u−2, u−1, v, v1, v2, . . .) shadows y (the dotindicates the position of the zeroth coordinate). Necessarily, f(y) = f [π(w)] =π[σ(w)] ∈ V s[vii≥1] ∩ Z(v1) = W s(f(x), Z(v1)). Thus f(y) ∈W s(f(x), Z(v1)).

Since y ∈ W s(x, Z(v0)) was arbitrary, f [W s(x, Z(v0))] ⊂ W s(f(x), Z(v1)). Theother inequality is symmetric.

Overlapping charts property: The following holds for all ε is small enough.Suppose Z,Z ′ ∈ Z and Z ∩ Z ′ 6= ∅.

(1) If Z = Z(Ψpu0 ,p

s0

x0 ), Z ′ = Z(Ψqu0 ,q

s0

y0 ), then Z ⊂ Ψy0([−(qu0 ∧ qs0), qu0 ∧ qs0]2).


(2) For all x ∈ Z, y ∈ Z ′, V u(x, Z) intersects V s(y, Z ′) at a unique point.(3) For any x ∈ Z ∩ Z ′, Wu(x, Z) ⊂ V u(x, Z ′) and W s(x, Z) ⊂ V s(x, Z ′).

Proof sketch. If Z ∩ Z ′ 6= ∅, then there are u, v ∈ Σ#(G ) s.t. u0 = Ψpu0 ,p

s0

x0 ,

v0 = Ψqu0 ,q

s0

y0 , and π(u) = π(v). By Theorem 4.5(4), Ψ−1y0Ψx0

is close to ±Id. Thisis enough to prove (1)–(3), see Lemmas 10.8 and 10.10 in [Sar13] for details.

Step 3 (Bowen, Sinai): A countable Markov partition. We refine Z into apartition without destroying the Markov property or the product structure.

The refinement procedure we use below is due to Bowen [Bow75], building onearlier work of Sinai [Sin68a, Sin68b]. It was designed for finite Markov covers, butworks equally well for locally finite infinite covers. Local finiteness is essential: Ageneral non-locally finite cover may not have a countable refining partition as canbe seen in the example of the cover (α, β) : α, β ∈ Q of R.

Enumerate Z = Zi : i ∈ N. For every Zi, Zj ∈ Z s.t. Zi ∩ Zj 6= ∅, let

Tusij := x ∈ Zi : Wu(x, Zi) ∩ Zj 6= ∅ , W s(x, Zi) ∩ Zj 6= ∅,Tu∅ij := x ∈ Zi : Wu(x, Zi) ∩ Zj 6= ∅ , W s(x, Zi) ∩ Zj = ∅,T∅sij := x ∈ Zi : Wu(x, Zi) ∩ Zj = ∅ , W s(x, Zi) ∩ Zj 6= ∅,

T∅∅ij := x ∈ Zi : Wu(x, Zi) ∩ Zj = ∅ , W s(x, Zi) ∩ Zj = ∅.

This is a partition of Zi. Let T :=Tαβij : i, j ∈ N, α ∈ u,∅, β ∈ s,∅

. This is

a countable set, and Tusii = Zi for all i, T ⊃ Z . Necessarily, T covers NUH#χ (f).

The Markov partition: R:= the collection of sets which can be put in the formR(x) :=

⋂T ∈ T : T 3 x for some x ∈

⋃i≥1 Zi.

Proposition 5.2. R is a countable pairwise disjoint cover of NUH#χ (f). It refines

Z , and every element of Z contains only finitely many elements of R.

Proof. See [Bow75] or [Sar13, Prop. 11.2]. The local finiteness of Z is needed toshow that R is countable: It implies that #T ∈ T : T 3 x <∞ for all x.

The following proposition says that R is a Markov partition in the sense of Sinai[Sin68b]. First, some definitions. The u–fibre and s–fibre of x ∈ R ∈ R are

Wu(x,R) :=⋂Wu(x, Zi) ∩ Tαβij : Tαβij ∈ T contains R

W s(x,R) :=⋂W s(x, Zi) ∩ Tαβij : Tαβij ∈ T contains R.

Proposition 5.3. The following properties hold:

(1) Product structure: Suppose R ∈ R.(a) If x ∈ R, then the s and u fibres of x contain x, and are contained in R,

therefore R =⋃x∈RW

u(x,R) and R =⋃x∈RW

s(x,R).(b) For all x, y ∈ R, either the u–fibres of x, y in R are equal, or they are

disjoint. Similarly for s–fibres.(c) For all x, y ∈ R, Wu(x,R) and W s(y,R) intersect at a unique point, de-

noted by [x, y] and called the Smale bracket of x, y.(2) Hyperbolicity: For all z1, z2 ∈ W s(x,R), distΛ(fn(z1), fn(z2)) −−−−→

n→∞0,

and for all z1, z2 ∈ Wu(x,R), distΛ(f−n(z1), f−n(z2)) −−−−→n→∞

0. The rates are

exponential.


(3) Markov property: Let R0, R1 ∈ R. If x ∈ R0 and f(x) ∈ R1, thenf [W s(x,R0)] ⊂W s(f(x), R1) and f−1[Wu(f(x), R1)] ⊂Wu(x,R0).

Proof. This follows from the Markov properties of Z as in [Bow75]. See [Sar13,Prop. 11.5–11.7] for a proof using the notation of this paper.

Step 4: Symbolic coding for f : Λ → Λ [AW67, Sin68b]. Let R denote thepartition we constructed in the previous section. Suppose R,S ∈ R. We say thatR connects to S, and write R → S, whenever ∃x ∈ R s.t. f(x) ∈ S. Equivalently,R→ S iff R ∩ f−1(S) 6= ∅.

The dynamical graph of R: This the directed graph G with set of vertices R

and set of edges (R,S) ∈ R ×R : R→ S.

Fundamental observation [AW67, Sin68b]: Suppose m ≤ n are integers, and

Rm → · · · → Rn is a finite path on G , then

`[Rm, . . . , Rn] := f−`(Rm) ∩ f−`−1(Rm+1) ∩ · · · ∩ f−`−(n−m)(Rn) 6= ∅.

Proof. This can be seen by induction on n − m as follows: If n − m = 0 or 1there is nothing to prove. Assume by induction that the statement holds for m−n,then ∃x ∈ `[Rm, . . . , Rn] and ∃y ∈ Rn s.t. f(y) ∈ Rn+1. Let z := [fn(x), y], thenf−n(z) ∈ `[Rm, . . . , Rn+1] by the Markov property.

The brevity of the proof should not hide the amazing nature of the statement.Here is a consequence: Given R0 ∈ R, if there is an x s.t. f i(x) ∈ Ri for i =−n, . . . , 0 and there is a y s.t. f i(y) ∈ Ri for i = 0, . . . , n, then there is a z s.t.f i(z) ∈ Ri for i = −n, . . . , n. Thus given the “present” R0, any concatenation ofa possible “past” R−n → · · · → R0 with a possible “future” R0 → · · · → Rn isrealized by some orbit. That such combinatorial independence of the near past fromthe near future (given the present)5 can be present in a deterministic dynamicalsystem is truly counterintuitive. It is a tool of immense power.

The sets `[Rm, . . . , Rn] can be related to cylinders in Σ#(G ) as follows. Define

for every path vm → · · · → vn on G (not G )

Z`(vm, . . . , vn) := π(u) : u ∈ Σ#(G ), ui = vi (i = `, . . . , `+ n−m).

Lemma 5.4. For every infinite path · · ·R0 → R1 → · · · on G there is a gpov ∈ Σ(G ) s.t. for every n, Rn ⊂ Z(vn) and −n[R−n, . . . , Rn] ⊂ Z−n(R−n, . . . , Rn).

Proof. See [Sar13, Lemma 12.2].

Proposition 5.5. Every vertex of G has finite outgoing and ingoing degrees.

Proof. Fix R0 ∈ R. For every path R−1 → R0 → R1 in G , find a path v−1 → v0 →v1 in G s.t. Z(vi) ⊃ Ri for |i| ≤ 1. Since Z is locally finite, there are finitely manypossibilities for v0. Since every vertex of G has finite degree, there are also onlyfinitely many possibilities for v−1, v1. Since every element in Z contains at mostfinitely many elements in R, there is a finite number of possibilities for R−1, R1.

5Notice the similarity to the Markov property from the theory of stochastic processes.


Let Σ(G ) := doubly infinite paths on G = R ∈ RZ : ∀i, Ri → Ri+1,equipped with the metric d(R,S) := exp[−min|i| : Ri 6= Si], and the action

of the left shift map σ : Σ(G )→ Σ(G ), σ(R)i = Ri+1. Let

Σ#(G ) :=R ∈ Σ(G ) : Rii≤0, Rii≥0 contain constant subsequences

.

Since π : Σ(G ) → Λ is Holder continuous, there are constants C > 0, θ ∈ (0, 1)s.t. for every finite path v−n → · · · → vn on G , diam(Z−n(v−n, . . . , vn)) ≤ Cθn. ByLemma 5.4, diam(−n[R−n, . . . , Rn]) ≤ Cθn for every finite path R−n → · · · → Rnon G . This allows us to make the following definition:

Symbolic dynamics for f : Let π : Σ(G )→ Λ be defined by

π(R) := The unique point in

∞⋂n=0

−n[R−n, . . . , Rn].

Theorem 5.6. The following holds for all ε small enough in the definition of gpos:

(1) π σ = f π.

(2) π : Σ(G )→ Λ is Holder continuous.

(3) π[Σ#(G )] has full µΛ–measure.

(4) Every x ∈ π[Σ#(G )] has finitely many pre-images in Σ#(G ). If µ is ergodic,this number is equal a.e. to a constant.

(5) Moreover, there exists N : R × R → N s.t. if x = π(R) where Ri = R forinfinitely many i < 0 and Ri = S for infinitely many i > 0, then #S ∈Σ#(G ) : π(S) = x ≤ N(R,S).

Proof. If R ∈ Σ(G ), then π(σ(R)) = f(π(R)):

π(σ(R)) =⋂n≥0

−n[R−n+1, . . . , Rn+1] ⊇⋂n≥0

−(n+2)[R−n−1, . . . , Rn+1]

!=⋂n≥0

f(−n−1[R−n−1, . . . , Rn+1]

) !⊃⋂n≥0

f(−n−1[R−n−1, . . . , Rn+1]

)!= f

⋂n≥0

−n−1[R−n−1, . . . , Rn+1]

= f(π(R)).

The equalities!= are because f is invertible. To justify

!⊃, it is enough to show that

f is continuous on an open neighborhood of C := −n−1[R−n−1, . . . , Rn+1]. Fix some

v0 = Ψpu0 ,p

s0

x0 s.t. Z(v0) ⊃ R0, then C ⊂ R0 ⊂ Z(v0) ⊂ Ψx0[(−Qε(x0), Qε(x0))2] ⊂

Λ \S. So f is continuous on C.

(2) is because of the inequality diam(−n[R−n, . . . , Rn]) ≤ Cθn mentioned above.

(3) is because for every x ∈ NUH#χ (f), x = π(R) where Ri := unique element

of R which contains f i(x). Clearly R ∈ Σ(G ). To see that R ∈ Σ#(G ), we useProposition 4.4 to construct a regular gpo v ∈ Σ#(G ) s.t. x = π(v). Necessarily,Ri ⊂ Z(vi). Every Z(v) contains at most finitely many elements of R (Proposition5.2). Therefore, the regularity of v implies the regularity of R.

(4) follows from (5) and the f–invariance of x 7→ #R ∈ Σ#(G ) : π(R) = x.


(5) is proved using Bowen’s method [Bow78, pp. 13–14], see also [PP90, p. 229].The proof is the same as in [Sar13], but since the presentation there has an error,we decided to include the complete details in the appendix.

Step 5: Symbolic dynamics for ϕ : M → M . Let π : Σ(G ) → Λ denote thesymbolic coding of f : Λ→ Λ given by Theorem 5.6.

Recall that R : Λ → (0,∞) denotes the roof function of Λ. By the choice ofΛ, R is bounded away from zero and infinity, and there is a global constant C s.t.supx∈Λ\S ‖dRx‖ < C, see Lemma 2.5. Let

r : Σ(G )→ (0,∞), r := R π.

This function is also bounded away from zero and infinity, and since π : Σ(G )→ Λis Holder and Pesin charts are connected subsets of Λ \S, r is Holder continuous.

Let σr : Σr → Σr denote the topological Markov flow with roof function r and

base map σ : Σ(G )→ Σ(G ) (see page 2 for definition). Recall that the regular part

of Σr is Σ#r := (x, t) : x ∈ Σ#(G ), 0 ≤ t < r(x). This is a flow invariant set,

which contains all the periodic orbits of σr. By Poincare’s recurrence theorem, Σ#r

has full measure with respect to every flow invariant probability measure. Let

πr : Σr →M , πr(x, t) := ϕt[π(x)].

The following claims follow directly from Theorem 5.6:

(1) πr σtr = ϕt πr for all t ∈ R.

(2) πr[Σ#r ] has full measure with respect to µ.

(3) Every p ∈ πr[Σ#r ] has finitely many pre-images in Σ#

r . In case µ is ergodic,

p 7→ #(π−1r (p) ∩ Σ#

r ) is ϕ–invariant, whence constant almost everywhere.(4) Moreover, there exists N : R × R → N s.t. if p = πr(x, t) where xi = R for

infinitely many i < 0 and xi = S for infinitely many i > 0, then #(y, s) ∈Σ#r : πr(y, s) = p ≤ N(R,S).

This proves all parts of Theorem 1.3, except for the Holder continuity of πr.

Step 6: Holder continuity of πr. Every topological Markov flow is continuouswith respect to a natural metric, introduced by Bowen and Walters. We will show

that πr : Σr →M is Holder continuous with respect to this metric.First we recall the definition of the Bowen-Walters metric. Let σr : Σr → Σr

denote a general topological Markov flow (cf. page 2).Suppose first that r ≡ 1 (constant suspension). Let ψ : Σ1 → Σ1 be the suspen-

sion flow, and make the following definitions [BW72]:

Horizontal segments: Ordered pairs [z, w]h ∈ Σ1 × Σ1 where z = (x, t) andw = (y, t) have the same height 0 ≤ t < 1. The length of a horizontal segment[z, w]h is defined to be `([z, w]h) := (1− t)d(x, y) + td(σ(x), σ(y)), where d is themetric on Σ given by d(x, y) := exp[−min|n| : xn 6= yn].

Vertical segments: Ordered pairs [z, w]v ∈ Σ1 × Σ1 where w = ψt(z) for some t.The length of a vertical segment [z, w]v is `([z, w]v) := min|t| > 0 : w = ψt(z). Basic paths from z to w: γ := (z0 = z

t0−→ z1t1−→ · · · tn−2−−−→ zn−1

tn−1−−−→ zn = w)with ti ∈ h, v such that [zi−1, zi]ti−1

is a horizontal segment when ti−1 = h ,

and a vertical segment when ti−1 = v. Define `(γ) :=∑n−1i=0 `([zi, zi+1]ti).


Bowen-Walters Metric on Σ1: d1(z, w) := inf`(γ) where γ ranges over all basicpaths from z to w.

Next we consider the general case r 6≡ 1. The idea is to use a canonical bijectionfrom Σr to Σ1, and declare that it is an isometry.

Bowen-Walters Metric on Σr [BW72]: dr(z, w) := d1(ϑr(z), ϑr(w)), whereϑr : Σr → Σ1 is the map ϑr(x, t) := (x, t/r(x)).

Lemma 5.7. Assume r is bounded away from zero and infinity, and Holder con-tinuous. Then dr is a metric, and there are constants C1, C2, C3 > 0, 0 < κ < 1which only depend on r such that for all z = (x, t), w = (y, s) in Σr:

(1) dr(z, w

)≤ C1[d(x, y)κ + |t− s|].

(2) Conversely:(a) If | t

r(x) −s

r(y) | ≤12 , then d(x, y) ≤ C2dr(z, w) and |s− t| ≤ C2dr(z, w)κ.

(b) If tr(x)−

sr(y) >

12 , then d(σ(x), y) ≤ C2dr(z, w) and |t−r(x)|, s ≤ C2dr(z, w).

(3) For all |τ | < 1, dr(στr (z), στr (w)) ≤ C3dr(z, w)κ.

See the appendix for a proof.

Lemma 5.8. πr : Σr →M is Holder with respect to the Bowen-Walters metric.

Proof. Fix (x, t), (y, s) ∈ Σr. If | tr(x) −

sr(y) | ≤ 1/2, then

distM (πr(x, t), πr(y, s)) = distM (ϕt(π(x)), ϕs(π(y)))

≤ distM (ϕt(π(x)), ϕs(π(x))) + distM (ϕs(π(x)), ϕs(π(y)))

≤ maxp∈M‖Xp‖ · |t− s|+ Lip(ϕs) Hol(π)d(x, y)δ

where Xp is the flow vector field of ϕ, and δ is the Holder exponent of π : Σ(G )→ Λ.The first summand is bounded by const dr(z, w)κ, by Lemma 5.7(2)(a). The

second summand is bounded by const dr(z, w)δ, because ϕ is a flow of a Lipschitz(even C1+β) vector field, therefore there are global constants a, b s.t. Lip(ϕs) ≤bea|s| [AMR88, Lemma 4.1.8], therefore Lip(ϕs) ≤ ba supR = O(1). It follows thatdistM (πr(x, t), πr(y, s)) ≤ const dr((x, t), (y, s))

minκ,δ.

Now assume that tr(x) −

sr(y) >

12 . Since ϕr(x)(π(x)) = π(σ(x)), we have

distM (πr(x, t), πr(y, s)) = distM (ϕt(π(x)), ϕs(π(y)))

≤ distM (ϕt(π(x)), ϕr(x)(π(x))) + distM (π(σ(x)), π(y)) + distM (π(y), ϕs(π(y)))

≤ maxp∈M‖Xp‖ · (|t− r(x)|+ |s|) + Hol(π)d(σ(x), y)δ ≤ const dr((x, t), (y, s))

δ,

By part (2)(b) of Lemma 5.7.In both cases we find that distM (πr(x, t), πr(y, s)) ≤ const dr((x, t), (y, s))

γ ,where γ := minκ, δ.

Part 3. Applications

6. Measures of maximal entropy

We use the symbolic coding of Theorem 1.3 to show that a geodesic flow on acompact smooth surface with positive topological entropy can have at most count-ably many ergodic measures of maximal entropy. This application requires dealingwith non-ergodic measures.


Lemma 6.1. Let ϕ be a continuous flow on a compact metric space X. If ϕ hasuncountably many ergodic measures of maximal entropy, then ϕ has at least onemeasure of maximal entropy with non-atomic ergodic decomposition.

Proof. Let Mϕ(X) denote the space of ϕ–invariant probability measures, togetherwith the weak star topology. This is a compact metrizable space [Wal82, Thm 6.4].The following claims are standard, but we could not find them in the literature:

Claim 1. Suppose E ⊂ X is Borel measurable, then µ 7→ µ(E) is Borel measurable.

Proof. Let M := E ⊂ X : E is Borel, and µ 7→ µ(E) is Borel measurable.Let A denote the collection of Borel sets E for which there are fn ∈ C(X) s.t.0 ≤ fn ≤ 1 and fn(x) −−−−→

n→∞1E(x) everywhere.

A is an algebra, because if 0 ≤ fn, gn ≤ 1 and fn → 1A, gn → 1B , then fngn →1A∩B , (1− fn)→ 1X\A, and (fn + gn − fngn) ∧ 1→ 1A∪B . A generates the Borel σ–algebra B(X), because it contains every open ballBr(x0): take fn(x) := ϕn[d(x, x0)] where ϕn ∈ C(R), 1[0,r− 1

n ] ≤ ϕn ≤ 1[0,r).

M ⊃ A : if A ∈ A , then by the dominated convergence theorem µ(A) =limn→∞

∫fndµ for the fn ∈ C(X) s.t. 0 ≤ fn ≤ 1 and fn → 1A. Since µ 7→

∫fndµ

is continuous, µ 7→ µ(A) is Borel measurable. M is closed under increasing unions and decreasing intersections.

By the monotone class theorem [Sri98, Prop. 3.1.14], M contains the σ–algebragenerated by A , whence M = B(X). The claim follows.

Claim 2. Eϕ(X) := µ ∈Mϕ(X) : µ is ergodic is a Borel subset of Mϕ(X).

Proof. Fix a countable dense collection fnn≥1 ⊂ C(X), 0 ≤ fn ≤ 1, then µ

is ergodic iff lim supk→∞∫| 1k∫ k

0fn ϕtdt −

∫fndµ|dµ = 0 for every n. This is a

countable collection of Borel conditions.

Claim 3. The entropy map µ 7→ hµ(ϕ) is Borel measurable.

Proof. Let T := ϕ1 (the time-one map of the flow ϕ), then hµ(ϕ) = hµ(T ). Thushµ(ϕ) = hµ(T ) = lim

n→∞hµ(T, αn) for any sequence of finite Borel partitions αn s.t.

maxdiam(A) : A ∈ αn −−−−→n→∞

0 [Wal82, Thm 8.3]. The claim follows, since it

easily follows from claim 1 that µ 7→ hµ(T, αn) is Borel measurable.

Let Emax(X) denote the set of ergodic measures with maximal entropy. Byclaims 2 and 3, this is a Borel subset of Mϕ(X). By the assumptions of the lemma,Emax(X) is uncountable. Every uncountable Borel subset of a compact metricspace carries a non-atomic Borel probability measure, because it contains a subsethomeomorphic to the Cantor set [Sri98, Thm 3.2.7]. Let ν be a non-atomic Borelprobability measure s.t. ν[Emax(X)] = 1, and let m :=

∫Emax(X)

µdν(µ). This is

a ϕ–invariant measure with non-atomic ergodic decomposition. Since the entropymap is affine [Wal82, Thm 8.4], m has maximal entropy.

Theorem 6.2. Suppose ϕ is a C1+β flow with positive speed and positive topologicalentropy on a C∞ compact three dimensional manifold, then ϕ has at most countablymany ergodic measures of maximal entropy.

Proof. Let h := topological entropy of ϕ, and assume by way of contradiction thatϕ has uncountably many ergodic measures of maximal entropy. By Lemma 6.1, ϕhas a measure of maximal entropy µ with a non-atomic ergodic decomposition.


By the variational principle [Wal82, Thm 8.3], hµ(ϕ) = h. By the affinity ofthe entropy map [Wal82, Thm 8.4], almost every ergodic component µx of µ hasentropy h. Fix some 0 < χ0 < h. By Ruelle’s entropy inequality [Rue78], a.e.ergodic component µx is χ0–hyperbolic. Consequently, µ is χ0–hyperbolic.

This places us in the setup considered in part 2, and allows us to apply Theorem1.3 to µ. We obtain a coding πr : Σr →M s.t. µ[π(Σ#

r )] = 1 and πr : Σ#r →M is

finite-to-one (though not necessarily bounded-to-one).

Lifting Procedure: Define a measure µ on Σr by setting for E ⊂ Σr Borel

µ(E) :=

∫πr(Σ#

r )

(1

|π−1r (p) ∩ Σ#

r |

∑πr(x,t)=p

1E(x, t)

)dµ(p), (6.1)

then µ is a σr–invariant measure, µ π−1r = µ, and hµ(σr) = hµ(ϕ).

Proof. We start by clearing away all the measurability concerns. Let X := Σr andY := M ]X (disjoint union). Define f : Σr → Y by f Σ#

r= πr and f Σr\Σ#

r= id,

then f : X → Y is a countable-to-one Borel map between polish spaces. Such mapssend Borel sets to Borel sets [Sri98, Thm 4.12.4], so πr(Σ

#r ) = f(Σ#

r ) is Borel.Next we show that the integrand in (6.1) is Borel. Let B := (x, f(x)) : f(x) ∈

M. This is a Borel subset of X×Y , because the graph of a Borel function is Borel[Sri98, Thm 4.5.2]. For every y ∈ Y , By := x ∈ X : (x, y) ∈ B is countable,because either y ∈ M and By := π−1

r (y) ∩ Σ#r , or y 6∈ M and then By = ∅. By

Lusin’s theorem [Sri98, Thm 5.8.11], there are countably many partially definedBorel functions ϕn : Mn → X s.t. B =

⋃∞n=1(ϕn(y), y) : y ∈ Mn. Write B =⊎∞

n=1(ϕn(y), y) : y ∈ M ′n, M ′n := y ∈ Mn : k < n, y ∈ Mk ⇒ ϕn(y) 6= ϕk(y).Then for every y ∈M ,

π−1r (y) = ϕn(y) : n ≥ 1, y ∈M ′n, and m 6= n⇒ ϕm(y) 6= ϕn(y).

Thus, the integrand in (6.1) equals∑∞n=1 1M ′n(p)1E(ϕn(p))

/∑∞n=1 1M ′n(p), a Borel

measurable function.Now that we know that (6.1) makes sense it is a trivial matter to see that it

defines a measure µ. This measure is σr–invariant because of the flow invarianceof µ and the commutation relation πr σr = ϕ πr. It has the same entropy as µ,because finite-to-one factor maps preserve entropy [AR62].

Projection Procedure: Every σr–invariant probability measure m on Σr projectsto a ϕ–invariant probability measure m := m π−1

r on M with the same entropy.

Proof. By Poincare’s recurrence theorem every σr–invariant probability measure iscarried by Σ#

r , therefore πr : (Σr, m)→ (M,m) is a finite-to-one factor map. Suchmaps preserve entropy.

Combining the lifting procedure and the projection procedure we see that thesupremum of the entropies of ϕ–invariant measures on M equals the supremum ofthe entropies of σr–invariant measures on Σr, and therefore µ given by (6.1) is ameasure of maximal entropy for σr.

Claim. σr has at most countably many ergodic measures of maximal entropy.

Proof. We recall the well-known relation between measures of maximal entropy forσr and equilibrium measures for the shift map σ : Σ→ Σ [BR75]: S := Σ× 0 isa Poincare section for σr : Σr → Σr, therefore every measure of maximal entropy

µ for σr can be put in the form µ =∫

Σ

∫ r(x)

0δ(x,t)dt dµΣ(x)

/∫ΣrdµΣ where µΣ


is a shift-invariant measure on Σ. The denominator is well-defined, because r isbounded away from zero and infinity. If µ is ergodic, µΣ is ergodic.

By Abramov’s formula, hµ(σr) = hµΣ(σ)/ ∫

ΣrdµΣ. Similar formulas hold for all

other σr–invariant probability measures m and the measures mΣ they induce onΣ. Since µ is a measure of maximal entropy, hmΣ

(σ)/ ∫

ΣrdmΣ = hm(σr) ≤ h (the

maximal possible entropy) for all σ–invariant measures mΣ. This is equivalent tosaying that hmΣ

(σ) +∫

Σ(−hr)dmΣ ≤ 0, with equality iff hm(σr) = h.

Thus, if µ is a measure of maximal entropy for σr, then µΣ is an equilibriummeasure for −hr, where h is the value of the maximal entropy. Also P (−hr) :=suphν(σ)− h

∫Σrdν = 0, where the supremum ranges over all σ–invariant prob-

ability measures ν on Σ.Recall that r : Σ → R is Holder continuous. By [BS03], a Holder continuous

potential on a topologically transitive countable Markov shift has at most oneequilibrium measure. If the condition of topological transitivity is dropped, thenthere are at most countably many such measures (see the proof of Thm 5.3 in[Sar13]). It follows that there are at most countably many possibilities for µΣ, andtherefore at most countably many possibilities for µ.

We can now obtain the contradiction which proves the theorem. Consider theergodic decomposition of µ given by (6.1). Almost every ergodic component isa measure of maximal entropy (because the entropy function is affine). By theclaim there are at most countably many different such measures. Therefore theergodic decomposition of µ is atomic: µ =

∑piµi with µi ergodic and pi ∈ (0, 1)

s.t.∑pi = 1. Projecting to M , and noting that factors of ergodic measures are

ergodic, we find that µ =∑piµi where µi := µi π−1

r are ergodic. This is anatomic ergodic decomposition for µ. But the ergodic decomposition is unique, andwe assumed that µ has a non-atomic ergodic decomposition.

7. Mixing for equilibrium measures on topological Markov flows

Let σr : Σr → Σr be a topological Markov flow, together with the Bowen-Waltersmetric (see Lemma 5.7). Let Φ : Σr → R be bounded and continuous.

The topological pressure of Φ: P (Φ) := suphµ(σr) +∫

Φdµ, where thesupremum ranges over all σr–invariant probability measures µ on Σr.

An equilibrium measure for Φ: A σr–invariant probability measure µ on Σrs.t. hµ(σr) +

∫Φdµ = P (Φ).

Theorem 7.1. Suppose µ is an equilibrium measure of a bounded Holder contin-uous potential for a topological Markov flow σr : Σr → Σr. If σr is topologicallytransitive, then the following are equivalent:

(1) if eiθr = h/h σ for some Holder continuous h : Σ→ S1 and θ ∈ R, then θ = 0and h = const,

(2) σr is weak mixing,(3) σr is mixing.

(3) ⇒ (2) ⇒ (1) are because if eiθr = h/h σ, then F (x, t) = e−iθth(x) is aneigenfunction of the flow. (1) ⇒ (2) ⇒ (3) are known in the special case when Σis a subshift of finite type: Parry and Pollicott proved (1)⇒(2) [PP90], and Ratnerproved (2)⇒(3)⇒Bernoulli [Rat74],[Rat78]. Dolgopyat showed us a different proofof (2)⇒(3) (private communication).


These proofs can be pushed through to the countable alphabet case with someeffort, using the thermodynamic formalism for countable Markov shifts [BS03]. Thedetails can be found in [LLS14].

The following theorem is a symbolic analogue of Plante’s necessary and sufficientcondition for a transitive Anosov flow to be a constant suspension of an Anosovdiffeomorphism [Pla72], see also [Bow73]:

Theorem 7.2. Let σr : Σr → Σr be a topologically transitive topological Markovflow. Either every equilibrium measure of a bounded Holder continuous potentialis mixing, or there is Σ′r ⊂ Σr of full measure s.t. σr : Σ′r → Σ′r is topologicallyconjugate to a topological Markov flow with constant roof function.

Proof. If Σ is a finite set, then Σr equals a single closed orbit, and the claim istrivial. From now on assume that Σ is infinite.

Assume σr is not mixing, then exp[iθr] = h/h σ with h : Σ → S1 Holdercontinuous and θ 6= 0. Write θ = 2π/c and put h in the form h = exp[iθU ], whereU : Σ→ R is Holder continuous. Necessarily r + U σ − U ∈ cZ.

We are free to change U on every partition set by a constant in cZ to make sureU is bounded and positive. Fix N > 2‖U‖∞/ inf(r).

Construction: There is a cylinder A =−m [y−m, . . . , yn] such that

(i) m,n > 0 and y−m = yn,(ii) nA(·) > N on A, where nA(x) := infn ≥ 1 : σn(x) ∈ A,

(iii) x, x′ ∈ A⇒ |U(x)− U(x′)| < N inf(r).

To find A, take y ∈ Σ with dense orbit. Since Σ is infinite, σk(y) are distinct.

Therefore, y has a cylindrical neighborhood C s.t. σk(C)∩C = ∅ for k = 1, . . . , N .Choose m,n > 0 so large that −m[y−m, . . . , yn] ⊂ C, and |U(x)−U(x′)| < N inf(r)for all x, x′ ∈ C. Since y has a dense orbit, every symbol appears in y infinitelyoften in the past and in the future, therefore we can choose m,n so that y−m = yn.The cylinder A = −m[y−m, . . . , yn] satisfies (i), (ii) and (iii), because A ⊂ C.

Since µ is ergodic and globally supported, the following set has full µ–measure:Σ′r := z ∈ Σr : σtr(z) ∈ A× 0 infinitely often in the past and in the future.

Step 1: σr : Σ′r → Σ′r is topologically conjugate to a topological Markov flowσr∗ : Σ∗r∗ → Σ∗r∗ whose roof function r∗ takes values in cZ.

Proof. A × 0 is a Poincare section for σr : Σ′r → Σ′r. The roof function isrA := r + r σ + · · ·+ r σnA−1. By (ii), inf(rA) > N inf(r), so 0 < U < inf(rA).

Let S := σU(x)r (x, 0) : (x, 0) ∈ Σ′r. This is a Poincare section for σr : Σ′r → Σ′r,

and its roof function is r∗A := rA + U σnA − U (this is always positive becauseU < inf(rA)). All the values of r∗A belong to cZ, as can be seen from the identity

r∗A =∑nA−1k=0 (r + U σ − U) σk.

We claim that the section map of S is topologically conjugate to a topologicalMarkov shift. Let V denote the collection of sets of the form

〈B〉 := σU(x)r (x, 0) : x ∈ −m[A,B,A],

where A = (y−m, . . . , yn) is the word defining A, and B is any other word s.t.

−m[A,B,A] 6= ∅ for which the only appearances of A in (A,B,A) are at thebeginning and the end.


It is easy to see that σU(x)r (x, 0) ∈ S iff x = (. . . , A,B1, A,B2, A, . . .) with

〈Bi〉 ∈ V , and that any sequence 〈Bi〉i∈Z ∈ V Z appears this way. Let π : S → V Z

be the map π(x) = 〈Bi〉i∈Z. Since A appears in (A,Bi, A) only at the beginning

and the end, π σr∗Ar = σ π, with σ = the left shift on V Z. So the section map of

S is topologically conjugate to the shift on V Z. Let Σ∗ := V Z.The roof function w.r.t. this new coding is r∗ := r∗A π−1. Routine calculations

show that the Holder continuity of r implies the Holder continuity of r∗. So σr∗ :Σ∗r∗ → Σ∗r∗ is a TMF, and σr : Σ′r → Σ′r is topologically conjugate to σr∗ .

Step 2: σr∗ : Σ∗r∗ → Σ∗r∗ is topologically conjugate to a topological Markov flow

σr : Σr → Σr where r takes values in cZ, and r(x) = r(x0).

Proof. Since r∗ is Holder continuous and takes values in cZ, there must be somen0 > 0 s.t. r∗ is constant on every cylinder of the form −n0 [a−n0 , . . . , an0 ].

Take π(x, t) := (xii∈Z, t), where xi := (x−n0+i, . . . , xn0+i). The reader can

check that the collection of xii∈Z thus obtained is a topological Markov shift Σ,and that r(xii∈Z) only depends on the first symbol x0.

Step 3: σr : Σr → Σr is topologically conjugate to a topological Markov flow

σr : Σr → Σr where r is constant equal to c.

Proof. The set (x, kc) : x ∈ Σ, k ∈ Z, 0 ≤ kc < value of r on 0[x0] is a Poincaresection for the suspension flow with constant roof function (equal to c). The section

map is conjugate to a topological Markov shift Σ which we now describe.

Let G = G (V , E) denote the graph of Σ. Then Σ = Σ(G ), where G has set of

vertices V := (vk

): v ∈ V , 0 ≤ kc < value of r on 0[v] and edges

(vk

)→(vk+1

)when

(vk+1

)∈ V , and

(vk

)→(u0

)when

(vk+1

)6∈ V and u → v is in E. The

conjugacy π : Σr → Σr is π(x, t) := (σbt/cc(y), t − bt/ccc), where y is given by

(. . . ;(x0

0

),(x0

1

), . . . ,

(x0

r(x0)/c−1

);(x1

0

),(x1

1

), . . . ,

(x1

r(x1)/c−1

); . . .) with

(x0

0

)at the zeroth

coordinate.

8. Counting simple closed orbits

Let π(T ) := #[γ] : `[γ] ≤ T, γ simple closed geodesic. In this section we provethe following generalization of Theorem 1.1:

Theorem 8.1. Suppose ϕ is a C1+β flow with positive speed and positive topologicalentropy h on a C∞ compact three dimensional manifold M . If ϕ has a measure ofmaximal entropy, then π(T ) ≥ C(ehT /T ) for all T large enough and C > 0.

This implies Theorem 1.1, because every C∞ flow admits a measure of maximalentropy. Indeed, by a theorem of Newhouse [New89], ϕ1 : M → M admits a

measure of maximal entropy m, and µ :=∫ 1

0m ϕtdt has maximal entropy for ϕ.

Discussion. The theorem strengthens Katok’s bound lim infT→∞1T log π(T ) ≥ h,

see [Kat80, Kat82] for general flows, and it improves Macarini and Schlenk’s boundlim infT→∞

1T log π(T ) > 0 for the class of Reeb flows in [MS11].

If one assumes more on the flow, then much better bounds for π(T ) are known:

(1) Geodesic flows on closed hyperbolic surfaces: π(T ) ∼ et/t [Hub59].


(2) Topologically weak mixing Anosov flows (e.g. geodesic flows on compact sur-faces with negative curvature): π(T ) ∼ CehT /T [Mar69]. C = 1/h (C. Toll,unpublished). See [PS98] for estimates of the error term. The earliest estimatesfor π(T ) in variable curvature are due to Sinai [Sin66].

(3) Topologically weak mixing Axiom A flows: π(T ) ∼ ehT /hT [PP83]. See [PS01]for an estimate of the error term.

(4) Geodesic flows on compact rank one manifolds: C1(ehT /T ) ≤ π0(T ) ≤ C2(ehT /T )for some C1, C2 > 0, where π0(T ) counts the homotopy classes of simple closedgeodesics with length less than T [Kni97, Kni02].

(5) Geodesic flows for certain non-round spheres: For certain metrics constructedby [Don88, BG89], π(T ) ∼ ehT /hT [Wea].

We cannot give upper bounds for π(T ) as in (1)–(5), because in the generalsetup we consider there can be compact invariant sets with lots of closed geodesicsbut zero topological entropy (e.g. embedded flat cylinders). Such sets have zeromeasure for any ergodic measure with positive entropy, and they lie outside the“sets of full measure” that we can control using the methods of this paper. Addingto our pessimism is the existence of Cr (1 < r < ∞) surface diffeomorphismswith super-exponential growth of periodic points [Kal00]. The suspension of theseexamples gives Cr flows with super-exponential growth of closed orbits. To thebest of our knowledge, the problem of doing this in C∞ is still open.

Preparations for the proof of Theorem 8.1. Fix an ergodic measure of max-imal entropy for ϕ, and apply Theorem 1.2 with this measure. The result is atopological Markov flow σr : Σr → Σr together with a Holder continuous mapπr : Σr →M , satisfying (1)–(6) in Theorem 1.2.

We saw in the proof of Theorem 6.2 (see page 33) that if ϕ has a measureof maximal entropy, then σr has a measure of maximal entropy. By the ergodicdecomposition, σr has an ergodic measure of maximal entropy. Fix such a measure

µ, and write µ =∫

Σ

∫ r(x)

0δ(x,t) dt dν(x)

/ ∫Σrdν. The induced measure ν is an

ergodic shift invariant measure on Σ. When we proved Theorem 6.2, we saw thatν is an equilibrium measure of φ = −hr.

Like all ergodic shift invariant measures, ν is supported on a topologically tran-sitive topological Markov shift Σ′ ⊆ Σ [ADU93]. There is no loss of generality inassuming that σ : Σ→ Σ is topologically transitive (otherwise work with Σ′).

Proof of Theorem 8.1 when µ is mixing. Fix 0 < ε < 10−1 inf(r). Since r isHolder, there are H > 0 and 0 < α < 1 s.t. |r(x)− r(y)| ≤ Hd(x, y)α. Recall that

d(x, y) = exp[−min|n| : xn 6= yn]. For every ` ≥ 1, if xn0+`−n0

= yn0+`−n0

then

|r`(x)− r`(y)| ≤`−1∑i=0

Hd(σi(x), σi(y))α ≤ H`−1∑i=0

e−αminn0+i,n0+`−i <2He−αn0

1− e−α·

Choose n0 s.t. sup|r`(x)− r`(y)| : xn0+`−n0

= yn0+`−n0

, ` ≥ 1 < ε. Fix some cylinder

A := −n0 [a−n0 , . . . , an0 ] s.t. ν(A) 6= 0, and let Υ(T ) :=⊎∞n=1 Υ(T, n), where

Υ(T, n) := (y, n) : y ∈ A, σn(y) = y, |rn(y)− T | < 2ε

Given (y, n) ∈ Υ(T, n), let γy,n : [0, rn(y)] → M , γy,n(t) = πr[σtr(y, 0)]. This is

a closed orbit with length `(γy,n) = rn(y) ∈ [T − 2ε, T + 2ε]. But γy,n(t) is not

necessarily simple, because π is not injective.


Let γsy,n := γy,n [0,`(γy,n)/N ], N = N(y, n) := #0 ≤ t < `(γy,n) : γy,n(t) =

γy,n(0). This is a simple closed orbit. N = 1 iff γy,n is simple, and N <

`(γy,n)/ inf(r), because an orbit with length less than inf(r) cannot be closed.

We obtain a map Θ : Υ(T )→ [γ]: γ is a simple closed s.t. `(γ) ≤ T + 2εΘ : (y, n) 7→ [γsy,n].

Θ is not one-to-one, but we there is a uniform bound on its non-injectivity:

1 ≤#Θ−1([γsy,n])

n≤ c0. (8.1)

Here is the proof. Suppose (y, n), (z,m) ∈ Υ(T ) and [γsy,n] = [γsz,m], then:

N(y, n) = N(z,m): [ T−2εN(y,n) ,

T+2εN(y,n) ] and [ T−2ε

N(z,m) ,T+2εN(z,m) ] both contain ` = `(γsy,n)

= `(γsz,m). But N(y, n), N(z,m) < T+2εinf(r) , and [T−2ε

j , T+2εj ] are pairwise disjoint

for j = 1, . . . , [T+2εinf(r) ], because ε < 1

10 inf(r).

[γy,n] = [γz,m], because [γsy,n] = [γsz,m] and N(y, n) = N(z,m).

n = m, because n, m are the number of times γy,n, γz,m enter Λ0 := πr(Σ×0),and equivalent closed orbits enter Λ0 the same number of times.

πr(z, 0) = πr(σk(y), 0) for some k = 0, . . . , n− 1, because πr(z, 0) ∈ γy,n ∩ Λ0.

y, z ∈ Σ#, and yi = a0 for infinitely many i < 0 and infinitely many i > 0.

By Theorem 1.2(5) there is a constant c0 := N(a0, a0) s.t. if xi = a0 for infinitelymany i > 0 and infinitely many i < 0, then #z ∈ Σ# : π(z) = π(x) ≤ c0. So

#Θ−1([γsy,n]) ≤ #[Σ#r ∩

⋃n−1k=0 π

−1r πr(σk(y), 0)] ≤ c0n. Also #Θ−1([γsy,n]) ≥ n,

because [γsσk(y),n] = [γsy,n] for k = 0, . . . , n− 1. This proves (8.1).

By (8.1) and the fact shown above that [γsy,n] = [γsz,m]⇒ m = n,

#[γ] : γ simple closed orbit s.t. `(γ) ≤ T + 2ε ≥

≥ #[γsy,n] : (y, n) ∈ Υ(T ) =

∞∑n=1

#[γsy,n] : (y, n) ∈ Υ(T, n)

∞∑n=1

#Υ(T, n)

n, where An Bn means ∃C,N0 s.t. ∀n > N0, C−1 ≤ An

Bn≤ C

=

∞∑n=1

1

n

∑σn(y)=y

1A(y)1[−2ε,2ε](rn(y)− T )

ehT

T

∞∑n=1

∑σn(y)=y

1A(y)1[−2ε,2ε](rn(y)− T )e−hrn(y)

=ehT

TS(T ), where S(T ) :=

∞∑n=1

∑σn(y)=y

1A(y)1[−2ε,2ε](rn(y)− T )e−hrn(y).

To prove the theorem, it is enough to show that lim inf S(T ) > 0.Recall that ν is an equilibrium measure of φ = −hr and P (−hr) = 0 (see the

claim on page 33). The structure of such measures was found in [BS03]. We willnot repeat the characterization here, but we will simply note that it implies thefollowing uniform estimate [BS03, page 1387]: There exists G(a) > 1 s.t. for every


cylinder of the form 0[b] = 0[ξ0, . . . , ξn] with ξn = a, C(a)−1 ≤ ν(0[b])exp(−hrn(y)) ≤

C(a) for all y ∈ 0[b]. It follows that there is a constant G = G(A) s.t.

G−1 ≤exp[−hrn(y)]

ν(−n0[a−n0

, . . . , a−1; y0, . . . , yn−1; a0, . . . , an0])≤ G

for every y ∈ −n0 [a−n0 , . . . , a−1; y0, . . . , yn−1; a0, . . . , an0 ]. Let

UT :=⋃

(y,n)∈Υ(T )

−n0[a−n0

, . . . , a−1; y0, . . . , yn−1; a0, . . . , an0],

then S(T ) ν[UT ] = (ε−1∫rdν)µ(UT × [0, ε]).

We claim that

UT × [0, ε] ⊃ (A× [0, ε]) ∩ σ−Tr (A× [0, ε]). (8.2)

Once this shown, we can use the mixing of µ to get lim inf µ(UT×[0, ε]) > 0, whencelim inf S(T ) > 0.

Suppose (x, t), σTr (x, t) ∈ A× [0, ε], and write σTr (x, t) = (σn(x), t+ T − rn(x)).Since x ∈ A ∩ σ−n(A), there exists y ∈ A s.t. σn(y) = y and yn+n0

−n0= xn+n0

−n0. By

the choice of n0, |rn(x)− rn(y)| < ε. Since t, t+ T − rn(x) ∈ [0, ε], |rn(x)− T | < ε,whence |rn(y)− T | < 2ε. So (x, t) ∈ UT × [0, ε]. This proves (8.2).

Proof of Theorem 8.1 when µ is not mixing. In this case, Theorem 7.1 givesus a set of full measure Σ′r ⊂ Σr s.t. σr : Σ′r → Σ′r is topologically conjugateto a constant suspension over a topologically transitive topological Markov shift

σc : Σ× [0, c)→ Σ× [0, c). Let ϑ : Σ× [0, c)→ Σ′r denote the topological conjugacy,

and let p : Σ× [0, c)→M be the map p := πr ϑ.The map p has the same finiteness-to-one properties of πr, because looking care-

fully at the proof of Theorem 7.1, we can see that if x ∈ Σ contains some symbolv infinitely many times in its future (resp. past), then ϑ(x, t) = (y, s) where ycontains some symbol a = a(v) infinitely many times in its future (resp. past).

Since σr : Σ′r → Σ′r has a measure of maximal entropy, σc : Σ× [0, c)→ Σ× [0, c)

has a measure of maximal entropy. By Abramov’s formula, σ : Σ → Σ has ameasure of maximal entropy, and the value of this entropy is hc.

Gurevich characterized the countable state topological Markov shifts which pos-sess measures of maximal entropy [Gur69, Gur70]. His work shows that there are

p ∈ N, C > 0, and a vertex v s.t. #x ∈ Σ : x0 = v, σnp(x) = x enp·hc.Such x determines a simple closed orbit γsx,np : [0, npc

N(x,np) ]→M , where γsx,np(t) =

p[σtc(x, 0)], N(x, np) := #0 ≤ t < npc : p[σtc(x, 0)] = p[(x, 0)]. We get a map

Θ : (x, n) : x0 = v, σnp(x) = x → [γ] : γ simple, `(γ) ≤ npc , Θ(x, n) := [γsx,np].

Θ is not one-to-one, but one can show as before that 1 ≤ #Θ−1([γsx,np])

np ≤ C(v), where

C(v) = N(a(v), a(v)). Thus #[γsy,np] : y ∈ Σ, y0 = v, σnp(y) = y enp·hc

n . Since

`(γsy,np) ≤ npc, #[γ] : `(γ) ≤ Tn, γ is simple ≥ const(ehTn/Tn) for Tn = npc. It

follows that #[γ] : `(γ) ≤ T, γ is simple ≥ const(ehT /T ) for large T .


Appendix A: Standard proofs

Proof of Lemma 2.1. M is compact and smooth, so there is a constant rinj > 0

s.t. for every p ∈M , expp : ~v ∈ TpM : ‖~v‖p ≤ rinj →M is√

2–bi-Lipschitz onto

its image (see e.g. [Spi79], chapter 9). Fix 0 < r < rinj, and complete ~np :=Xp‖Xp‖

to an orthonormal basis ~np, ~up, ~vp of TpM . Then

Jp(x, y) := expp(x~up + y~vp)

is a C∞ diffeomorphism from Ur := (x, y) ∈ R2 : x2 + y2 ≤ r onto Sr(p) for all0 < r < rinj, proving that S = Sr(p) is a C∞ embedded disc.

We claim that distM (·, ·) ≤ distS(·, ·) ≤ 2 distM (·, ·). The first inequality isobvious. For the second, suppose z1, z2 ∈ S. There are ~v1, ~v2 ⊥ Xp s.t. ‖~vi‖p ≤ rand zi = expp(~vi). Let γ(t) := expp[t~v2+(1−t)~v1], t ∈ [0, 1]. Clearly, γ ⊂ S, whence

distS(z1, z2) ≤ length(γ). Since expp has bi-Lipschitz constant√

2, distS(z1, z2) ≤√2‖~v1 − ~v2‖p ≤ (

√2)2 distM (z1, z2).

We bound ](Xq, TqS) for q ∈ S. If ~u,~v, ~w ∈ R3 \ 0, then |](~u, span~v, ~w)| ≥| sin](~u, span~v, ~w)| ≥

∣∣∣ 〈~u,~v,~w〉‖~u‖·‖~v‖·‖~w‖

∣∣∣ , where 〈~u,~v, ~w〉 is the signed volume of the

parallelepiped with sides ~u,~v, ~w. So for every q ∈ Sr(p),

|](Xq, TqS)| ≥ A(p, x) :=

∣∣∣∣∣∣∣⟨XJp(x), (dJp)x

∂∂x , (dJp)x

∂∂y

⟩Jp(x)

‖XJp(x)‖Jp(x) · ‖(dJp)x ∂∂x‖Jp(x) · ‖(dJp)x ∂

∂y‖Jp(x)

∣∣∣∣∣∣∣ ,where x = x(q) is characterized by q = Jp(x). By definition A(p, 0) = 1, so thereis an open neighborhood Vp of p and a positive number δp s.t. A(q, x) > 1

2 onWp := Vp ×Bδp(0).

Working in M×R3, we cover K := M×0 by a finite collection Wp1, . . . ,WpN ,

and let rleb be a Lebesgue number. Then A(p, x) > 12 for every p ∈ M and

‖x‖ < rleb. The lemma follows with rs := 12 min1, rinj, rleb.

Uniform Inverse Function Theorem. Let F : U → V be a differentiable mapbetween two open subsets of Rd s.t. det(dFx) 6= 0 for all x ∈ U . Suppose there

are K,H, β s.t. ‖dFx‖, ‖(dFx)−1‖ ≤ K and ‖dFx1− dFx2

‖ ≤ H‖x1 − x2‖β for all

x1, x2 ∈ U . If x ∈ U , Bε(x) ⊂ U , and 0 < ε < 2−(β+1)/β(KH)−1/β, then

(1) F−1 is a well-defined differentiable open map on W := Bδ(F (x)), δ := ε/(2K),(2) ‖(dF−1)y

1− (dF−1)y

2‖ ≤ H∗‖y

1− y

2‖β for all y

1, y

2∈W , with H∗ := K3H.

Proof. Track the constants in the fixed point theorem proof of the inverse functiontheorem (see e.g. [Sma74]).

Proof of Lemma 2.2. Let B := x ∈ R3 : ‖x‖ < 1. Let V be a finite open coverof M such that for every V ∈ V ,

(1) V = CV (B) where CV : B → V is a C2 diffeomorphism,(2) CV extends to a bi-Lipschitz C2 map from a neighborhood of B onto V ,(3) (dCV )x

∂∂x , (dCV )x

∂∂y , XCV (x) are linearly independent for x ∈ B.

Since M is compact and X has no zeroes, ‖Xp‖ is bounded from below. This,together with the C1+β regularity of X, implies that ~np := Xp/‖Xp‖ is Lipschitz

on M . Apply the Gram-Schmidt procedure to ~nCV (x), (dCV )x∂∂x , (dCV )x

∂∂y for

x ∈ B. The result is a Lipschitz orthonormal frame ~np, ~up, ~vp for TpM , p ∈ V .


Define for every p ∈ V the function Fp(x, y, t) := ϕt[expp(x~up + y~vp)]. (dFp)0 is

non-singular for every p ∈ V . Since p 7→ det(dFp)0 is continuous and V is compact,

det(dFp)0 is bounded away from zero for p ∈ V . Since (p, x, y, t) 7→ det(dFp)(x,y,t)

is uniformly continuous on V × B, ∃δ(V ) > 0 s.t. det(dFp)(x,y,t) is bounded away

from zero on (p, x, y, t) : p ∈ V , x2 + y2 ≤ δ(V )2, |t| ≤ δ(V ).Fix 0 < δ < minδ(V ) : V ∈ V s.t. δ < rleb/2S0 where S0 := 1+maxp∈M ‖Xp‖

and rleb is a Lebesgue number for V . For every p ∈ M , Fp((x, y, t) : x2 + y2 ≤

δ2, |t| ≤ δ)⊂ Brleb

(p), so ∃V ∈ V s.t. Fp((x, y, t) : x2 + y2 ≤ δ2, |t| ≤ δ

)⊂ V =

dom(C−1V ). For this V ,

G = Gp,V := C−1V Fp : (x, y, t) : x2 + y2 ≤ δ2, |t| ≤ δ → R3

is a well-defined map, with Jacobian uniformly bounded away from zero. A directcalculation shows that ‖dG(x,y,t)‖, ‖(dG(x,y,t))

−1‖ and the β–Holder norm of dGare uniformly bounded by constants that do not depend on p, V .

By the uniform inverse function theorem, for every 0 < δ′ ≤ δ, the imageG((x, y, t) : x2 + y2 ≤ (δ′)2, |t| ≤ δ′

)contains a ball B∗ of some fixed radius

d′(δ′) centered at C−1V (p), and G can be inverted on B∗. So F−1

p is well-definedand smooth on CV (B∗). Since CV is bi-Lipschitz, there is a constant d(V, δ′) s.t.CV (B∗) ⊃ Bd(V,δ′)(p), so F−1

p is well-defined and smooth on Bd(V,δ′)(p). The C1+β

norm of the F−1p there is uniformly bounded by a constant which only depends

on V . It follows that (q, t) 7→ ϕt(q) can be inverted with bounded C1+β norm onBd(V,δ′)(p).

Let K(V ) denote a bound on the Lipschitz constant of the inverse function, andlet ρ(V ) := δ/2K(V ), then (q, t) 7→ ϕt(q) is a diffeomorphism from Sρ(V )(p) ×[−ρ(V ), ρ(V )] onto FBρ(V )(p). Let rf := minρ(V ) : V ∈ V , then (q, t) 7→ ϕt(q)is a diffeomorphism from Srf (q)× [−rf , rf ] onto FBrf (p). The lemma follows with

this rf , and with d := mind(V, 12 rf ) : V ∈ V .

Proof of Lemma 2.3. We use the notation of the previous proof. Invert thefunction Fp(x, y, t) := ϕt[expp(x~up + y~vp)] on Bd(p):

F−1p (z) = (xp(z), yp(z), tp(z)) (z ∈ Bd(p)).

By the uniform inverse function theorem, the C1+β norm ofG−1 is bounded by someconstant independent of p, V . Since F−1

p = G−1 C−1V , CV is bi-Lipschitz, and V

is finite, xp(·), yp(·), tp(·) have uniformly bounded Lipschitz constants (independentof p), and the differentials of xp, yp, tp are β–Holder with uniformly bounded Holderconstants (independent of p).

Clearly tp(z) := tp(z) and qp(z) := expp[xp(z)~up + yp(z)~vp] are the unique so-

lutions for z = ϕtp(z)[qp(z)]. Thus tp, qp are Lipschitz functions with Lipschitzconstant bounded by some L independent of p, and C1+β norm bounded by someH independent of p.

Proof of Theorem 5.6(5). The proof is motivated by [Bow78].Say that R,R′ ∈ R are affiliated, if there are Z,Z ′ ∈ Z s.t. R ⊂ Z, R′ ⊂ Z ′,

and Z ∩ Z ′ 6= ∅. Let N(R,S) := N(R)N(S), where

N(R) := #(R′, v′) ∈ R ×A : R′ is affiliated to R and Z(v′) ⊃ R′.

This is finite, because of the local finiteness of Z .


Let x = π(R) where Ri = R for infinitely many i < 0 and Ri = S for infinitelymany i > 0. Let N := N(R,S), and suppose by way of contradiction that x

has N + 1 different pre-images R(0), . . . , R(N) ∈ Σ#(G ), with R(0) = R. Write

R(j) = R(j)k k∈Z. By Lemma 5.4 there are v(j) ∈ Σ(G ) s.t. for every n,

−n[R(j)−n, . . . , R

(j)n ] ⊂ Z−n(v

(j)−n, . . . , v

(j)n ) and R(j)

n ⊂ Z(v(j)n ).

For every j, v(j) ∈ Σ#(G ), because R(j) ∈ Σ#(G ) and Z is locally finite. It follows

that π(v(j)) ∈ Z−n(v(j)−n, . . . , v

(j)n ) for all n.6

Since x = π(R(j)) ∈ −n[R−n, . . . , Rn] ⊂ Z−n(v(j)−n, . . . , v

(j)n ). Since the diame-

ter of Z−n(v(j)−n, . . . , v

(j)n ) tends to zero as n → ∞ by the Holder continuity of π,

π(v(j)) = x.

It follows that Z(v(0)i ), . . . , Z(v

(N)i ) all intersect (they contain f i(x) = π(σiv(j))).

Since R(j)i ⊂ Z(v

(j)i ) and Z(v

(0)i ), . . . , Z(v

(N)i ) intersect, R

(0)i , . . . , R

(N)i are affiliated

for all i.

In particular, if k, ` > 0 satisfy R(0)−k = R and R

(0)` = S (there are infinitely many

such k, `), then there are at most N = N(R)N(S) possibilities for the quadruple

(R(j)−k, Z(v

(j)−k);R

(j)` , Z(v

(j)` )), j = 0, . . . , N . By the pigeonhole principle, there are

0 ≤ j1, j2 ≤ N s.t. j1 6= j2 and

(R(j1)−k , v

(j1)−k ) = (R

(j2)−k , v

(j2)−k ) and (R

(j1)` , v

(j1)` ) = (R

(j2)` , v

(j2)` ).

We can also guarantee that

(R(j1)−k , . . . , R

(j1)` ) 6= (R

(j2)−k , . . . , R

(j2)` ).

To do this fix in advance some m s.t. (R(j)−m, . . . , R

(j)m ) (j = 0, . . . , N) are all

different, and work with k, ` > m.

Now let A := R(j1), B := R(j2), a := v(j1), b := v(j2). Write A−k = B−k =: B,A` = B` =: A, a−k = b−k =: b, and a` = b` =: a. Choose

xA ∈ −k[A−k, . . . , A`] and xB ∈ −k[B−k, . . . , B`].

Define two points zA, zB by the equations

f−k(zA) := [f−k(xB), f−k(xA)] ∈Wu(f−k(xB), B) ∩W s(f−k(xA), B)

f `(zB) := [f `(xB), f `(xA)] ∈Wu(f `(xB), A) ∩W s(f `(xA), A).

This makes sense, because f−k(xA), f−k(xB) ∈ B and f `(xA), f `(xB) ∈ A.One checks using the Markov property of R that zA ∈ −k[A−k, . . . , A`], and

zB ∈ −k[B−k, . . . , B`]. Since (A−k, . . . , A`) 6= (B−k, . . . , B`) and the elements of Rare pairwise disjoint, zA 6= zB . We will obtain the contradiction we are after byshowing that zA = zB .

Since f `(zA) ∈ A` = A ⊂ Z(a) and f−k(zB) ∈ B−k = B ⊂ Z(b), there areα, β ∈ Σ#(G ) s.t. zA = π(α), zB = π(β), α` = a, β−k = b. Let c = cii∈Z whereci = βi for i ≤ −k, ci = ai for −k < i < `, and ci = αi for i ≥ `. This belongs toΣ#(G ), because α, β ∈ Σ#(G ) and β−k = b = a−k and a` = a = α`. We will show

that zA = π(c) = zB . Write ci = Ψpui ,p

si

xi (i ∈ Z).

6At this point the proof given in [Sar13] has a mistake. There it is claimed that π(v(j)) ∈Z−n(v

(j)−n, . . . , v

(j)n ) without making the assumption that R(j) ∈ Σ#(G ).


By the definition of zA, zB and the Markov property, f−k(zA), f−k(zB) bothbelong to Wu(f−k(xB), B). It follows that

Wu(f−k(zA), B) = Wu(f−k(zB), B) = Wu(π(σ−kβ), B) ⊂ V u[(ci)i≤−k].

It follows that f i(zA), f i(zB) ∈ Ψxi([−Qε(xi), Qε(xi)]2) for all i ≤ −k.Similarly, f `(zA), f `(zB) both belong to W s(f `(xA), A), whence

W s(f `(zA), A) = W s(f `(zB), A) = W s(π(σ`α), A) ⊂ V s[(ci)i≥`].

It follows that f i(zA), f i(zB) ∈ Ψxi([−Qε(xi), Qε(xi)]2) for all i ≥ `.For −k < i < `, f i(zA), f i(zB) ∈ Ai ∪Bi ⊂ Z(ai) ∪ Z(bi). The sets Z(ai), Z(bi)

intersect, because as we saw above

x = π(a) ∈ Z−k(a−k, . . . , a`), whence f i(x) ∈ Z(ai), x = π(b) ∈ Z−k(b−k, . . . , b`), whence f i(x) ∈ Z(bi).

By the overlapping charts property of Z (see §5) and since ai = ci for −k < i < `,

Z(ai) ∪ Z(bi) ⊂ Ψxi([−Qε(xi), Qε(xi)]2) (i = −k + 1, . . . , `− 1).

In summary, f i(zA), f i(zB) ∈ Ψxi([−Qε(xi), Qε(xi)]2) for all i ∈ Z. As shown inthe proof of the shadowing theorem (Thm. 4.2), c must shadow zA and zB , whencezA = zB .

Remark. We take this opportunity to correct a mistake in [Sar13]. Theorem 12.8in [Sar13] (the analogue of the statement we just proved) is stated wrongly as a

bound for the number of all pre-images of x ∈ π[Σ#(G )]. But what is actuallyproved there (and all that is needed for the remainder of the paper) is just a bound

on the number of pre-images which belong to Σ#(G ) (denoted there by Σ#χ ).

Thus the statements of Theorems 1.3 and 1.4 in [Sar13] should be read as bounds

on the number of pre-images in Σ#χ (denoted here by Σ#(G )), and not as bounds

on the number of pre-images in Σχ (denoted here by Σ(G )). The other results orproofs in [Sar13] are not affected by these changes, since Σχ \Σ#

χ does not contain

any periodic orbits, and because Σχ \Σ#χ has zero measure for every shift invariant

probability measure (Poincare’s recurrence theorem).

Proof of Lemma 5.7. Let ψ : Σ1 → Σ1 denote the constant suspension flow, then

For every horizontal segment [z, w]h, |τ | < 1 =⇒∣∣∣ `([ψτ (z),ψτ (w)]h)

`([z,w]h) − 1∣∣∣ ≤ 2e2|τ |.

This uses the trivial bound d(x, y)/d(σk(x), σk(y)) ∈ [e−1, e] for |k| ≤ 1 and themetric d(x, y) := exp[−min|n| : xn 6= yn].

For every vertical segment [z, w]v, `([ψτ (z), ψτ (w)]v) = `([z, w]v) for all τ .

Thus for all z, w ∈ Σ1, |τ | < 1 =⇒ (1 + 2e2|τ |)−1 ≤ d1(ψτ (z),ψτ (w))d1(z,w) ≤ (1 + 2e2|τ |).

Claim: dr is a metric on Σr.

Proof. It is enough to show that d1 is a metric. Symmetry and the triangle in-equality are obvious; we show that d1(z, w) = 0⇒ z = w.

Let z = (x, t), w = (y, s), τ := 12 − t. If d1(z, w) = 0, then d1(ψτ (z), ψτ (w)) = 0.

Let γ = (z0, z1, . . . , zn) be a basic path from ψτ (z) to ψτ (w) with length less thanε, with ε < 1

3 fixed but arbitrarily small. Write zi = (xi, ti), then ψτ (z) = (x0, t0)and ψτ (w) = (xn, tn).


Since the lengths of the vertical segments of γ add up to less than ε and t0 = 12 ,

γ does not leave Σ× [ 12−ε,

12 +ε]. It follows that |tn−t0| < ε. Since ε was arbitrary,

tn = t0, and ψτ (z), ψτ (w) have the same second coordinate.Since γ does not leave Σ × [ 1

2 − ε,12 + ε], it does not cross Σ × 0. Writing

a list of the horizontal segments [(xik , tik), (xik+1, tik+1)]h, we find that xik+1 =

xik+1. By the triangle inequality ε > d1(ψτ (x, t), ψτ (y, t)) ≥ e−1

∑d(xik , xik+1

) ≥e−1d(x0, xn). Since ε is arbitrary, x0 = xn, and ψτ (z), ψτ (w) have the same firstcoordinate. Thus ψτ (z) = ψτ (w), whence z = w.

Part (1): dr((x, t), (y, s)) ≤ const[d(x, y)α + |t − s|], where α denotes the Holderexponent of r.

Proof. dr((x, t), (y, s)) ≡ d1((x, tr(x) ), (y, s

r(y) )). The basic path (x, tr(x) ), (x, s

r(y) ),

(y, sr(y) ) shows that d1((x, t

r(x) ), (y, sr(y) )) ≤ | t

r(x) −s

r(y) |+ ed(x, y) ≤ 1inf(r)

[|t− s|+

Holα(r)d(x, y)α]

+ ed(x, y) ≤ const[d(x, y)α + |t− s|].

Part (2): Let α denote the Holder exponent of r. There is a constant C2 whichonly depends on r s.t. for all z = (x, t), w = (y, s) in Σr:

(a) If∣∣ tr(x) −

sr(y)

∣∣ ≤ 12 , then d(x, y) ≤ C2dr(z, w) and |s− t| ≤ C2dr(z, w)α.

(b) If tr(x) −

sr(y) >

12 , then d(σ(x), y) ≤ C2dr(z, w) and |t− r(x)|, s ≤ C2dr(z, w).

Proof. These estimates are trivial when dr(z, w) is bounded away from zero, so itis enough to prove part (2) for z, w s.t. dr(z, w) < ε0, with ε0 a positive constantthat will be chosen later.

Suppose∣∣ tr(x) −

sr(y)

∣∣ < 12 and let τ := 1

2 −t

r(x) (a number in (− 12 ,

12 ]), then

dr(z, w) = d1(ϑr(z), ϑr(w)) ≥ (1 + 2e2|τ |)−1d1(ψτ [ϑr(z)], ψτ [ϑr(w)])

≥ (1 + 2e2)−1d1((x, 12 ), (y, 1

2 + δ)), where δ := sr(y) −

tr(x) .

Notice that (y, 12 + δ) ∈ Σ1, because |δ| < 1

2 .

Suppose ε0(1 + 2e2) < 14 , then d1((x, 1

2 ), (y, 12 + δ)) < 1

4 . The basic paths whose

lengths approximate d1((x, 12 ), (y, 1

2 + δ)) are not long enough to leave Σ × [ 14 ,

34 ],

and they cannot cross Σ×0. For such paths the lengths of the vertical segmentsadd up to at least δ, and the lengths of the horizontal segments add up to at leaste−1d(x, y). Since d1((x, 1

2 ), (y, 12 + δ)) ≤ (1 + 2e2)dr(z, w),

d(x, y) ≤ e(1 + 2e2)dr(z, w) and |δ| ≤ (1 + 2e2)dr(z, w).

In particular, d(x, y) ≤ const dr(z, w), and |s−t| =∣∣r(y) s

r(y)−r(x) tr(x)

∣∣ ≤ sup(r)|δ|+|r(y)− r(x)| ≤ (1 + 2e2) sup(r)dr(z, w) + Holα(r)d(x, y)α ≤ const dr(z, w)α, wherethe last inequality uses our estimate for d(x, y) and the finite diameter of dr.

This proves part (a) when∣∣∣ tr(x) −

sr(y)

∣∣∣ < 12 . If t

r(x) −s

r(y) = 12 , repeat the

previous argument with τ := 0.49− tr(x) .

For part (b), suppose tr(x) −

sr(y) >

12 , and let τ := r(x)−t

r(x) + 12 . Now ψτ [ϑr(z)] =

(σ(x), 12 ) and ψτ [ϑr(w)] = (y, 1

2 + δ′), where δ′ := 1−(

tr(x) −

sr(y)

). As before

d(σ(x), y) ≤ e(1 + 2e2)dr(z, w) and |δ′| ≤ (1 + 2e2)dr(z, w).

Using s ≤ r(y)δ′, |r(x)−t| ≤ r(x)δ′, we see that s, |t−r(x)| < (1+2e2) sup(r)dr(z, w).


Part (3). There are constants C3 > 0, 0 < κ < 1 which only depend on r s.t. forall z, w ∈ Σr and |τ | < 1, dr(σ

τr (z), στr (w)) ≤ C3dr(z, w)κ.

Proof. We will only discuss the case τ > 0. The case τ < 0 can be handled

similarly, or deduced from the following symmetry: Let Σ := x : x ∈ Σ where

xi := x−i, and let r(x) := r(σx) (a function on Σ). Then Θ(x, t) = (σx, r(x)− t) is

a bi-Lipschitz map from Σr to Σr, and Θ σ−τr = στr Θ. This symmetry reflectsthe representation of the flow σ−tr with respect to the Poincare section Σ× 0.

We will construct a constant C ′3 s.t. for all z, w ∈ Σr, if 0 < τ < 12 inf(r), then

dr(στr (z), στr (w)) ≤ C ′3dr(z, w)α. Part (3) follows with κ := αN , C3 := (C ′3)

11−α ,

N := d1/min1, 12 inf(r)e.

We will also limit ourselves to the case when C2dr(z, w) < 12 inf(r); part (3) is

trivial when dr(z, w) is bounded away from zero.Let z := (x, t), w := (y, s). Since τ > 0, στr (z) = (σm(x), εr(σm(x)) and στr (w) =

(σn(y), ηr(σn(y)) where 0 ≤ ε, η < 1 and m,n ≥ 0. Notice that m,n ∈ 0, 1,(because 0 < τ < 1

2 inf(r), so σtr(z), σtr(w) cannot cross Σ× 0 twice).

Case 1:∣∣∣ tr(x) −

sr(y)

∣∣∣ ≤ 12 and m = n. In this case,

dr(στr (z), στr (w)) = d1((σm(x), ε), (σm(y), η))

≤ d1((σm(x), ε), (σm(y), ε)) + d1((σm(y), ε), (σm(y), η))

≤ ed(σm(x), σm(y)) + |ε− η|≤ e2d(x, y) + |ε− η| ≤ e2C2dr(z, w) + |ε− η|, by part (2)(a).

Since m = n, |ε− η| =∣∣∣ t+τ−rm(x)r(σm(x)) −

s+τ−rm(y)

r(σm(y))

∣∣∣ ≤ 1inf(r)2 [I1 + I2 + I3], where:

I1 = |tr(σm(y)) − sr(σm(x))| ≤ t|r(σm(x)) − r(σm(y))| + |t − s|r(σm(x)) ≤sup(r)[eαCα2 Holα(r) + C2]dr(z, w)α by part (2)(a).

I2 = τ |r(σm(x))− r(σm(y))| ≤ eαCα2 inf(r) Holα(r)dr(z, w)α (because m ≤ 1). I3 = |rm(x)r(σm(y)) − rm(y)r(σm(x))| ≤ |rm(x) − rm(y)|r(σm(y)) + rm(y) ·|r(σm(x))− r(σm(y))| ≤ const Holα(r)dr(z, w)α (because m ≤ 1).

Thus |ε − η| ≤ const dr(z, w)α, where the constant only depends on r. It followsthat dr(σ

τr (z), στr (w)) ≤ const dr(z, w)α where the constant only depends on r.

Case 2:∣∣∣ tr(x) −

sr(y)

∣∣∣ ≤ 12 and m 6= n. We can assume that n = m+ 1, thus

dr(στr (z), στr (w)) = d1((σm(x), ε), (σm+1(y), η))

≤ d1((σm(x), ε), (σm(y), ε)) + d1((σm(y), ε), (σm+1(y), η))

≤ ed(σm(x), σm(y)) + 1− ε+ η

≤ e2C2dr(z, w) + 1− ε+ η, by part (2)(a).

In our scenario, t+ τ − rm+1(x) is negative, and s+ τ − rm+1(y) is non-negative.The distance between these two numbers is bounded by |t−s|+|rm+1(x)−rm+1(y)|,whence by const dr(z, w)α. So |t+τ−rm+1(x)|, |s+τ−rm+1(y)| ≤ const dr(z, w)α.

Since 1 − ε = |t+τ−rm+1(x)|r(σmx) , η =

s+τ−rm+1(y)

r(σm+1y) , and the denominators are at least

inf(r), there is a constant which only depends on r s.t. 1 − ε, η < const dr(z, w)α.It follows that dr(σ

τr (z), στr (w)) ≤ const dr(z, w)α.


Case 3: tr(x) −

sr(y) >

12 and m = n. We have

dr(στr (z), στr (w)) = d1((σm(x), ε), (σm(y), η))

≤ d1((σm(x), ε), (σm+1(x), η)) + d1((σm+1(x), η), (σm(y), η))

≤ 1− ε+ η + e2C2dr(z, w), by part (2)(b), and since m ≤ 1.

Because t+ τ − rm+1(x) < 0 ≤ s+ τ − rm(y), it follows by part (2)(b) that

|t+ τ − rm+1(x)|, |s+ τ − rm(y)| ≤ |t− s− rm+1(x)− rm(y)|≤ |t− r(x)|+ s+ |rm(σ(x))− rm(y)| ≤ 2C2dr(z, w) + const dr(z, w)α.

As in case 2, this means that dr(στr (z), στr (w)) < const dr(z, w)α.

Case 4: tr(x) −

sr(y) >

12 and m 6= n. Recall that C2dr(z, w), τ < 1

2 inf(r). By part

(2)(b), s ≤ 12 inf(r), whence s+ τ < inf(r). Necessarily, n = 0, m = 1, m = n+ 1.

dr(στr (z), στr (w)) = d1((σn+1(x), ε), (σn(y), η))

≤ d1((σn+1(x), ε), (σn+1(x), η)) + d1((σn+1(x), η), (σn(y), η))

≤ |ε− η|+ ed(σ(x), y) (∵ n = 0)

≤ |ε− η|+ eC2dr(z, w), by part (2)(b).

We have |ε− η| =∣∣ t+τ−r(x)

r(σx) − s+τr(y)

∣∣ ≤ 1inf(r)2 [I1 + I2], where by part (2)(b)

I1 := |[t− r(x)]r(y)− sr(σx)| ≤ 2 sup(r)C2dr(z, w),

I2 := τ |r(σx)− r(y)| ≤ 12 inf(r) Hol(r)Cα2 dr(z, w)α.

It follows that dr(στr (z), στr (w)) ≤ const dr(z, w)α where the constant only depends

on r. This completes the proof of part (3).

9. Acknowledgements

The authors thank Anatole Katok, Francois Ledrappier and Federico Rodriguez-Hertz for useful and inspiring discussions. Special thanks go to Dmitry Dolgopyatfor showing us a proof that weak mixing ⇒ mixing in Theorem 7.1, GiovanniForni for suggesting the application to Reeb flows, and Edriss Titi for giving us anelementary proof that C1+ε vector fields generate flows with C1+ε time t maps.

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Department of Mathematics, University of Maryland at College Park, CollegePark, MD 20740, USA

E-mail address: [email protected]

Faculty of Mathematics and Computer Science, The Weizmann Institute of Science,

POB 26, Rehovot, Israel

E-mail address: [email protected]

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